6-1 Graphing Systems of Equations / 3 days
6-2 Substitution / 3 days
6-3 Adding and Subtracting / 3 days
6.1 - 6.3 Quiz / 1 days
6-4 Multiplication / 4 days
6-5 Graphing Systems of Inequalities / 3 days
Test Review / 1 day
Test / 1 day
Cumulative Review / 1 day
Total days in Unit 6 –Systems = 20 days
Review Question
What makes an equation linear? Exponents on variables are 1
What is a solution to a linear equation? Point (2, 7). Notice this point “works” in y = 3x +1.
Discussion
What do you think a system of equations is? 2 or more equations
What is a solution to a system of equations? Point that works in all equations
y = 2x – 3
y = -3x + 7
Notice that (2, 1) works in both equations.
What would that look like? Two lines
(2, 1) is where the two lines would intersect.
How many ways can two lines intersect?
# of Intersections / # of Solutions / How?1 / 1 / Different m’s
0 / 0 / Same m’s;
Different intercepts
Infinite / Infinite / Same m’s;
Different intercepts
SWBAT find the solution to a system of equations by graphing
Example 1: Graph each line to find the solution.
y = 2x – 3
y = -3x + 7
(2, 1) is the solution because that is the intersection point.
How do you know that your answer is correct? (2, 1) “works” in both equations
Example 2: Graph each line to find the solution.
y = 4x – 3
y = -3x + 7
The solution looks like it would be (2, 2).
How do you know that your answer is incorrect? (2, 2) doesn’t “work” in either equation
The correct solution is (10/7, 19/7).
Hmmmm?!?
What issue do you see with graphing to find the solution? It is not exact.
Example 3: Graph each line to find the solution.
y = 2x – 3
y = 2x + 3
The solution is No Solution.
What does the answer of No Solution mean? No points will work in both equations.
Example 4: Graph each line to find the solution.
y = 2x + 1
2y – 4x = 2
The solution is Infinite Solutions.
What does the answer of infinite solutions mean? There are an infinite amount of points that will work in both equations.
Can someone give me one of the possible answers? (0, 1) (1, 3)…
You Try!
Graph each line to estimate the solution.
1. y = 2x + 3(.5, .5)2. y = 4x – 1No Solution
y = -3x + 1 y – 4x = 2
3. y = 4x + 1Infinite Solutions4. y + 3x = 1 (1, -3)
3y = 12x + 3
______
5. 3y – 8 = 4x (2, 5)6. y = 5No Solution
x = 2 y = -2
What did we learn today?
Graph each line to estimate the solution.
1. y = 3x + 32. y = -3x + 4
y = -2x + 2 y = -3x + 2
3. y = 4 4. y + 3x = 1
y = 6
______
5. y = -2x – 16. y – 4 = 2x
x = 1 y = 2x + 6
7. y – 5x = 2 8. 4y = 3x – 2
y = 5x + 2 y = -2x – 2
9. y = 3x + 110. x = -3
4y = 12x + 4 x = 3
Review Question
What does a solution to a system of equations look like? Use your arms as lines to demonstrate each possibility. Point,Infinite,No Solution
Discussion
Can you look at a system of equations and tell whether it will have 1, infinite, or no solution? How? Yes, look at the slopes. Different Slopes: 1 solution, same slopes: no solution, same equation: infinite
SWBATfind the solution to a system of equations by graphing
Example 1: How many solutions? Estimate the solution by graphing.
y = 2x + 3
y = 2x + 5
No Solution; Same slopes
Example 2: How many solutions? Estimate the solution by graphing.
y = -3x + 2
y = 2x + 5
1; Different slopes, (-1, 3)
Example 3: How many solutions? Estimate the solution by graphing.
y = 2x + 5
4y – 8x = 20
Infinite Solutions; Same equations
You Try!
How many solutions? Then estimate the solution by graphing.
1. y = -4x + 11; Different slopes2. 3y = 2x + 3No Solution; Same slopes
y = 2x – 3 (1, -2)
______
3. x + 2y = 31; Different slopes4. y = 2x – 3Infinite Solutions; Same equation
3x – y = -5(-3, -3) 4x = 2y + 6
What did we learn today?
State how many solutions there are going to be then graph each line to find the solution.
1. y = 4x + 12. y = -2x + 5
y = -2x + 1 y + 2x = 2
3. x = 2 4. y + 2x = 1
x = 3
______
5. 2y + 4x = -26. y = 3x – 1
y = -2x – 1 -3x = y + 2
7. y – 2x = 2 8. 3y = 4x – 2
y = 2x + 2 y = -2x – 2
9. y = x + 110. y = 4
2y = -2x + 2 x = -1
Review Question
What are the three possibilities for a solution to a system of equations? Use your arms as lines to demonstrate each possibility. Point,Infinite, No Solution
Discussion
What is the major issue with solving a system of equations by graphing? It is not precise.
Today, we will be using the graphing calculator to find exact solutions.
SWBATfind the solution to a system of equations by using a graphing calculator
Example 1: How many solutions? 1
y = 4x – 2
y = -2x + 3
Graph to find the solution. (1, 1)
How do you know that your answer is wrong? That point does not work in both equations.
Let’s find the exact answer using the graphing calculator.
1. Press the “y =” button. Enter each equation.
2. Press graph.
3. Press 2nd, then trace.
4. Scroll down to 5: Intersect. Press enter.
5. Press enter 3 times.
(.83, 1.33)
Example 2: How many solutions? 0; Empty Set
y = -2x + 3
y + 2x = -5
Let’s confirm our answer using the graphing calculator.
(See instructions from above.)
Example 3: How many solutions? Infinite Solutions
y = 4x + 5
2y – 8x = 10
Let’s confirm our answer using the graphing calculator.
(See instructions from above.)
Example 4: How many solutions? 1
y = 8x – 1
3y + 5x = 55
Let’s find the exact answer using the graphing calculator.
(See instructions from above.)
Why can’t we see the intersection point? We have to change the window.
The solution is (2, 15).
What did we learn today?
Estimate the answer by graphing. Then find the exact answer using the graphing calculator.
1
1
1. y = 4x – 3
y = -2x + 2
3. 5y – 4x = 5
y = -2x + 2
2. y = 2x + 6
y– 2x = 1
4. 3y = 6x + 9
y = 2x + 3
1
Use the graphing calculator to find the exact answer. Then sketch the graph.
1
5. y = -3x + 2
y = 2x – 1
7. y = 5x – 1
y = 5x + 2
9. y – 4x = 3
y = 4x + 3
6. 5y = 3x – 2
y = -5x – 2
8. y = x + 1
3y = 3x + 3
10.
1
Review Question
What issue do we have with graphing? It isn’t exact.
Today we will discuss a way to find the exact answer to a system of equations.
Discussion
Solve: 2x + 5 = 11.
How can you check to make sure that ‘3’ is the correct answer? Substitute ‘3’ in for x.
What does substitution mean? Replacing something with something else.
That is what we will be doing today. This allows us to find exact answers to systems of equations. Since graphing did not.
Solving 2x + 5 = 11 is pretty easy.
Why would solving the following system be difficult?
y = 3x + 5
2x + 4y = 8
There are two equations and two variables. If we could get it down to one equation/one variable, it would be easy. This is what substitution allows us to do.
SWBATfind the solution to a system of equations by using substitution
Example 1: y = 3x – 2
2x + 3y = 27
We need to get rid of one variable/equation. We do this by substitution.
What is ‘y’ equal to? y = 3x – 2
So we substitute 3x – 2 in for ‘y’ in the second equation. When we do this, the second equation becomes:
2x + 3(3x – 2) = 27. Now solve. Notice how substitution got rid of an equation and a variable.
2x + 9x – 6 = 27
11x – 6 = 27
11x = 33
x = 3
Now substitute the ‘x’ back into either of the first two equations to get the ‘y’ value.
y = 3(3) – 2
y = 9 – 2
y = 7
The final answer is (3, 7).
What does the answer (3, 7) mean? That is the point of intersection.
Example 2: y = 3x + 5
6x – 4y = -32
We need to get rid of one variable/equation. We do this by substitution.
What is ‘y’ equal to? y = 3x + 5
So we substitute 3x + 5 in for ‘y’ in the second equation. When we do this, the second equation becomes:
6x – 4(3x + 5) = -32. Now solve. Notice how substitution got rid of an equation and a variable.
6x – 12x – 20 = -32
-6x – 20 = -32
-6x = -12
x = 2
Now substitute the ‘x’ back into either of the first two equations to get the ‘y’ value.
y = 3(2) + 5
y = 6 + 5
y = 11
The final answer is (2, 11).
What does the answer (2, 11) mean? That is the point of intersection.
Example 3: x + 5y = -3
3x – 2y = 8
What is different about this problem? None of the variables are solved for already
What variable should we solve for? Why? The ‘x’ in the first equation. It is the easiest.
When we solve for the ‘x’ in the first equation, we get: x = -3 – 5y.
So we substitute -3 – 5y in for ‘x’ in the second equation. When we do this, the second equation becomes: 3(-3 – 5y) – 2y = 8. Now solve. Notice how substitution got rid of an equation and a variable.
-9 – 15y – 2y = 8
-9 – 17y = 8
-17y =17
y = -1
Now substitute the ‘y’ back into the equation where we already solved for ‘x’ to get the ‘x’ value.
x = -3 – 5(-1)
x = -3 + 5
x = 2
The final answer is (2, -1).
What does the answer (2, -1) mean? That is the point of intersection.
Summarize
When is it easy to use substitution? When a variable is solved for or can be easily solved for
You Try!
1. y = 4x + 1 (3, 13)
3x + 2y = 35
2. x = 3y – 4 (2, 2)
2x + 4y = 12
3. 8x – 2y = 2(1, 3)
3x + y = 6
4. 2x – y = -4 (13, 30)
-3x + y = -9
What did we learn today?
Solve each system of equations using substitution. Confirm your answer by graphing.
1
1. y = 5x(2, 10)2. y = 3x + 4 (2, 10)
x + y = 12 3x + 2y = 26
3. x = 4y – 5 (-9, -1)4. y = 3x + 2 (3, 11)
2x + 3y = -21 2x + y = 17
Solve each system of equations using substitution.
5. y = 5x + 1(-2, -9)6. y = 2x + 2 (3, 8)
3x + y = -15 2x – 4y = -26
7. 3x + 2y = 7 (1, 2)8. y = 4x – 2 (2, 6)
x + 3y = 7 x – 3y = -16
9. 4x + y = 16 (3, 4)10. y = 5x (1, 5)
2x + 3y = 18 y = 3x + 2
1
Review Question
What are the three possibilities for a solution to a system of equations? Use your arms as lines to demonstrate each possibility. Point,Infinite, No Solution
Yesterday, we just addressed the first case.
Discussion
Solve: 2x + 5 = 2x + 7.
What does 5 = 7 mean? There is no solution to this problem.
Solve: 2x + 7 = 2x + 7.
What does 7 = 7 mean? There are infinite solutions to this problem.
SWBATfind the solution to a system of equations by using substitution
Example 1: y = 2x + 3
3x + 3y = 45
We need to get rid of one variable/equation. We do this by substitution.
What is ‘y’ equal to? y = 2x + 3
So we substitute 2x + 3 in for ‘y’ in the second equation. When we do this, the second equation becomes:
3x + 3(2x + 3) = 45. Now solve. Notice how substitution got rid of an equation and a variable.
3x + 6x + 9 = 45
9x + 9 = 45
9x = 36
x = 4
Now substitute the ‘x’ back into either of the first two equations to get the ‘y’ value.
y = 2(4) + 3
y = 8 + 3
y = 11
The final answer is (4, 11).
What does the answer (4, 11) mean? That is the point of intersection.
Example 2: y = 2x + 3
-4x + 2y = 6
We need to get rid of one variable/equation. We do this by substitution.
What is ‘y’ equal to? y = 2x + 3
So we substitute 2x + 3 in for ‘y’ in the second equation. When we do this, the second equation becomes:
-4x + 2(2x + 3) = 6. Now solve. Notice how substitution got rid of an equation and a variable.
-4x + 4x + 6 = 6
6 = 6
When does 6 = 6? Always
What does that mean? We have infinite solutions
What kind of lines do we have? They are the same line. (On top of each other.)
Example 3: -x + y = 4
-3x + 3y = 10
We need to get rid of one variable/equation. We do this by substitution.
What variable should we solve for? Why? ‘y’ in the first equation. It is the easiest.
When we solve for the ‘y’ in the first equation, we get: y = 4 + x.
So we substitute 4 + x in for ‘y’ in the second equation. When we do this, the second equation becomes:
-3x + 3(4 + x) = 10. Now solve. Notice how substitution got rid of an equation and a variable.
-3x + 12+ 3x = 10
12 = 10
When does 12 = 10? Never
What does that mean? There is no solution.
What kind of lines do we have? Parallel
Summarize
When is it easy to use substitution? When a variable is solved for or can easily be solved for
You Try!
1. y = 3x + 5(3, 14)
4x + 2y = 40
2. y = 2x + 3No Solution
-4x + 2y = 12
3. y – 3x = 4Infinite Solutions
-9x + 3y = 12
4. 4x + y = 11(2, 3)
3x – 2y = 0
What did we learn today?
Solve each system of equations using substitution. Confirm your answer by graphing.
1. y = 3x(3, 9)2. y = 4x + 3(2, 11)
2x + 3y = 33 3x + 2y = 28
Solve each system of equations using substitution.
3. y = 3x + 2 No Solution4. 4x + y = 13(3, 1)
-3x + y = 10 3x + 5y = 14
5. y = 5x + 2 Infinite Solutions6. 3x – 2y = 4(2, 1)
-10x + 2y = 4 -4x + y = -7
7. y = 4x– 3 (4, 13)8. 2x + y = 10Infinite Solutions
3x + 3y = 51 6x + 3y = 30
9. y = 3x + 1(3, 10)10. -2x + 8y = 8No Solution
2x + 3y = 36 x – 4y = 10
Review Question
When is it easy to use substitution? When a variable is solved for or can be easily solved for
Discussion
How do you get better at something? Practice
Therefore, we are going to practice solving systems of equations today.
We are going to have many days like this during the school year. In order for you to be successful, you need to take advantage of the time and ask questions from your classmates and teachers.
SWBATfind the solution to a system of equations by using substitution
Example 1: Let’s make sure we know how to use substitution.
y = 2x + 3
2x + 3y = 25
We need to get rid of one variable/equation. We do this by substitution.
What is ‘y’ equal to? y = 2x + 3
So we substitute 2x + 3 in for ‘y’ in the second equation. When we do this, the second equation becomes:
2x + 3(2x + 3) = 25. Now solve. Notice how substitution got rid of an equation and a variable.
2x + 6x + 9 = 25
8x + 9 = 25
8x = 16
x = 2
Now substitute the ‘x’ back into either of the first two equations to get the ‘y’ value.
y = 2(2) + 3
y = 4 + 3
y = 7
The final answer is (2, 7).
What does the answer (2, 7) mean? That is the point of intersection.
Let’s graph to confirm our answer.
You Try!
1. y = 5 – 2x(1, 3)2. y = 4 – 2x (1, 2)
3y + 3x = 12 2x – y = 0
3. x + y = 6No Solution4. 2x + y = 3Infinite Solutions
3x + 3y = 3 4x + 2y = 6
What did we learn today?
Solve each system of equations using substitution. Confirm your answer by graphing.
1. y = 3x(-1, -3)2. y = 3(1, 3)
2x + 3y = -11 3x+ 2y = 9
3. x= -4(-4, 2)4. 2x + y = 5No Solution
3x + y = -10 4x + 2y = 2
5. y = 2x + 1 (1, 3)6. 6x – 2y = 5Infinite Solutions
4x + 2y = 10 -12x + 4y = -10
Solve each system of equations using substitution.
7. y = 3x (-3, -9)8. x + 5y = 11(1, 2)
x + 2y = -21 3x – 2y = -1
9. y = 3x + 4(2, 10)10. -2x + 2y = 4(1, 3)
2x + 3y = 34 x – 4y = -11
11. y = 4x – 6 (3, 6)12. 2x + y = 7(3, 1)
3x + 4y = 33 3x – 2y = 7
13. y = 3x + 1(-2, -5)14. x + 3y = 14(5, 3)
2x + 3y = -19 2x – 4y = -2
15. x = 2y + 4(8, 2)16. -3x + 2y = -8(2, -1)
2x + 3y = 22 x – 4y = 6
Review Question
What are the three possibilities for a solution to a system of equations? Use your arms as lines to demonstrate each possibility. Point Infinite No Solution
Discussion
When is it easy to use substitution? When a variable is solved for or can be easily solved for
Why do we substitute something in for a variable?It allows us to get rid of one variable/equation.
How does this help us? We can solve one equation with one variable
Why wouldn’t substitution be good for the following system?When you solve for one of the
4x + 5y = 12variables, the result will be a fraction
4x – 3y = -4
What is something else that we could do? Subtract; it would get rid of one variable/equation.
(Remember our goal is to get rid of one variable/equation)
Remember the section title;
Remember the Alamo!)
Why are we allowed to add or subtract two equations to each other?
Since both sides are equal to each other, we can add/subtract to both sides.
Just like: 2x + 5 = 11
- 5 -5
So, when is it good to use addition/subtraction? When the coefficients are the same
How do you know whether to add or subtract? Same signs: subtract; Different signs: add
SWBATfind the solution to a system of equations by using addition/subtraction
Example 1: 3x – 2y = 4 Add the second equation to the first one.
4x + 2y = 10
7x = 14
x = 2
Now substitute the ‘x’ back into either of the first two equations to get the ‘y’ value.
3(2) – 2y = 4
6 – 2y = 4
-2y = -2
y = 1
The final answer is (2, 1).
What does the answer (2, 1) mean? That is the point of intersection.
Why would we use addition not subtraction? Because it eliminates the y’s
Example 2: 4x + 5y = 12 Subtract the second equation from the first one.
4x – 3y = -4
8y = 16
y = 2
Now substitute the ‘y’ back into either of the first two equations to get the ‘x’ value.
4x + 5(2) = 12
4x + 10 = 12
4x = 2
x = .5
The final answer is (.5, 2).
What does the answer (.5, 2) mean? That is the point of intersection.
Why would we use subtraction not addition? Because it eliminates the x’s
Example 3: 2x – 3y = 10(-1, -4)
2x = y + 2
What is different about this system? The x’s and y’s are not on the same side of the equation.
After a little bit of Algebra, we get the following system:
2x – 3y = 10Subtract the second equation from the first one. (-1, -4)
2x – y = 2
-2y = 9
y = -4
Now substitute the ‘y’ back into either of the first two equations to get the ‘x’ value.
2x – 3(-4) = 10
2x + 12 = 10
2x = -2
x = -1
The final answer is (-1, -4).
What does the answer (-4, -1) mean? That is the point of intersection.
Why would we use subtraction not addition? Because it eliminates the x’s
Summarize
When is it easy to use addition/subtraction? When the coefficients are the same
How do you know whether to add or subtract? Same signs: subtract; Different signs: add
You Try!
1. 4x – 5y = 10(5, 2)2. 3x + 5y = -16 (-2, -2)
2x + 5y = 20 3x – 2y = -2
3. y = 4x + 2(1, 6)4. -6x + 2y = 2(0, 1)
3x + 4y = 27 6x = 3y – 3
5. 4x + 2y = 16No Solution6. 3x + y = 16(5, 1)
4x + 2y = 10 6x – 3y =27
What did we learn today?
Use addition, subtraction, or substitution to solve each of the following systems of equations.
1. 3x + 2y = 22 (6, 2)2. 3x + 2y = 30 (4, 9)
3x – 2y = 14 y = 2x + 1
3. 3x – 5y = -35 (-5, 4) 4. 5x + 2y = 12 (2, 1)
2x – 5y = -30 -5x + 4y = -6
5. 4x = 7 – 5y (.5, 1)6. x = 6y + 11 (23, 2)
8x = 9 – 5y 2x + 3y = 52
7. x – 3y = 7 (4, -1)8. 3x + 5y = 12 Infinite Solutions
x + 2y = 2 3x + 5y = 12
9. 4x + y = 12 (2, 4)10. 2x + 3y = 5 (4, -1)
3x + 3y = 18 5x + 4y = 16
Review Question
When is it easy to use substitution? When a variable is solved for or can be easily solve for
When is it easy to use addition/subtraction? When the coefficients are the same
Discussion
What method should we use for problem #10 on the homework? Substitution
Why does this stink? It involves fractions.
How do you get better at something? Practice
Therefore, we are going to practice solving systems of equations today.
We are going to have many days like this during the school year. In order for you to be successful, you need to take advantage of the time and ask questions from your classmates and teachers.
SWBATfind the solution to a system of equations by using addition/subtraction
Example 1: 5x – 4y = 8 Add the second equation to the first one.
4x + 4y = 28
9x = 36
x = 4
Now substitute the ‘x’ back into either of the first two equations to get the ‘y’ value.
5(4) – 4y = 8
20 – 4y = 8
-4y = -12
y = 3
The final answer is (4, 3).
What does the answer (4, 3) mean? That is the point of intersection.
Why would we use addition not subtraction? Because iteliminates the y’s
Example 2: 5x + 5y = -5
5x – 3y = 11
8y = -16
y = -2
Now substitute the ‘y’ back into either of the first two equations to get the ‘x’ value.
5x + 5(-2) = -5
5x – 10 = -5
5x = -5
x = -1
The final answer is (-1, -2).
What does the answer (-1, -2) mean? That is the point of intersection.
Why would we use addition not subtraction? Because iteliminates the x’s
You Try!
1. 4x – 7y = -13(2, 3)
2x + 7y = 25
2. 3x + 4y = -9 (1, -3)
3x = 2y + 9
3. y = -2x – 3Infinite Solutions
4x + 2y = -6
4. 3x + 2y = 11No Solution
3x + 2y = 8
What did we learn today?
Use addition, subtraction, or substitution to solve each of the following systems of equations.
1. 5x + 4y = 14 (2, 1) 2. 3x + 6y = 21 (1, 3)
5x + 2y = 12 -3x+ 4y = 9
3. 5x + 2y = 6 (4, -7)4. y = -3x + 2 (0, 2)
9x + 2y = 22 3x + 2y = 4
5. 2x – 3y = -11 (-1, 3)6. 6x + 5y = 8 No Solution
x + 3y = 8 6x + 5y = - 2
7. x = 3y + 7 Infinite Solutions8. 3x – 4y = -5 (1, 2)
3x – 9y = 21 3x = -2y + 7
9. 2x + 3y = 1 (-1, 1)10. 4x – 5y= 2 (3, 2)
x + 5y = 4 6x + 5y = 28
Review Question
When is it easy to use substitution? When a variable is solved for or can be easily solved for
When is it easy to use addition/subtraction? When the coefficients are the same
Discussion
What is our goal when we are trying to solve a system of equations? Get rid of one variable
How does this help us? We can solve an equation with one variable.
SWBATsolve a word problem that involves a system of equations
Example 1: Find two numbers whose sum is 64 and difference is 42.
x + y = 64Add the second equation to the first one.
x – y = 42
2x = 106
x = 53
Now substitute the ‘x’ back into either of the first two equations to get the ‘y’ value.
53 + y = 64
y = 11
The two numbers are 53 and 11.
Example 2: Cable costs $50 for installation and $100/month. Satellite costs $200 for installation and $70/month. What month will the cost be the same?
C = 50 + 100mSubtract the second equation from the first one.
C = 200 + 70m
0 = -150 + 30m
150 = 30m
5 = m
What does 5 months represent? The month where it costs the same for both gyms.
How could this help you decide on which company to go with? Depending on how long you are going to keep your cable.
What did we learn today?
Use addition, subtraction, or substitution to solve each of the following systems of equations. Graph to confirm your answer in problem #1.
1. 2x + 2y = -2 (2, -3)2. 4x – 2y = -1 (-1, -1.5)
3x – 2y = 12 -4x + 4y = -2
3. 6x + 5y = 4 (-1, 2)4. x = 3y + 7 (4, -1)
6x – 7y = - 20 3x + 4y = 8
5. 2x – 3y = 12 (6, 0)6. 3x + 2y = 10 No Solution
4x + 3y = 24 3x + 2y = -8
7. -4x – 2y = -10 Infinite Solutions8. 8x+y = 10 (1, 2)
2x + y = 5 2x – 5y = -8
Write a system of equations. Then solve.
9. The sum of two numbers is 70 and their difference is 24. Find the two numbers. 23, 47
10. Twice one number added to another number is 18. Four times the first number minus the other number is 12. Find the numbers. 5, 8
11. Two angles are supplementary. The measure of one angle is 10 more than three times the other. Find the measure of each angle. 42.5, 137.5
12. Johnny is older than Jimmy. The difference of their ages is 12 and the sum of their ages is 50. Find the age of each person. 31, 19
13. The sum of the digits of a two digit number is 12. The difference of the digits is 2. Find the number if the units digit is larger than the tens digit. 5 and 7
14. A store sells Cd’s and Dvd’s. The Cd’s cost $4 and the Dvd’s cost $7. The store sold a total of 272 items and took in $1694. How many of each was sold? 202, 70
Review Question
When is it easy to use substitution? When a variable is solved for or can be easily solved for
When is it easy to use addition/subtraction? When the coefficients are the same
Discussion
What is our goal when we are trying to solve a system of equations? Get rid of one variable
How does this help us? We can solve an equation with one variable.
What method would you use to solve the following system of equations?
2x + 3y = 5
6x + 4y = 16
Why wouldn’t substitution be good? It would involve fractions.
Why wouldn’t add/subtract be good? It will not eliminate any of the variables
We need something else.
What could we do to the first equation to make it so we could subtract? Multiply by 3
Why are we allowed to do this? You are allowed to multiply the entire equation by whatever you want.
SWBATsolve a system of equations using multiplication
Example 1: 9x + 8y = 10 Multitpy by 2, then subtract18x + 16y = 20