Review #5: Solving a System of Linear Equations
A system of equations: two or more linear equations.
Solution to a system: the ordered pair that satisfies both equations and where they cross on a graph.
Methods to Solve a System of Linear Equations
Graphically / 1. Graph each linear equation.2. Determine the ordered pair of intersection.
3. Check the point in each linear equation.
Substitution
/ 1. Solve one of the equations for one of its variable.
2. Substitute the expression from step 1 into the other equation and solve for the other variable.
3. Substitute the value from step 2 into either original equation and solve to find the value of the variable is step 1.
Elimination / 1. Look for opposites in either variable
2. Add to the equations vertically to eliminate one variable.
3. Solve the resulting equation for the other variable.
4. Substitute the value into either original equation to find the value of the eliminated variable.
Solving a System of Equations Graphically:
Example 1:
Solve the following system graphically
y = 2x + 1
-x+ y = 7
Exercise 1:
Solve the following systems graphically
x+y =1
y= 14x-4
y = -x -2
2y =-2x+6
Extended Response
A business rents in-line skates and bicycles. During one day, the business has a total of 25 rentals and collects $450 for the rentals. It costs $15 per day to rent skates, and $30 per day to rent bicycles.
Step 1: Write a linear system.
Let x = skates, y = bicycles
Step 2: Graph the system
Step 3: Determine the number of skates rented and bicycles rented.
Multiple Choice Practice
Solving a System of Equations using Substitution
Directions:
1. Solve one of the equations for one of its variable
2. Substitute the expression from step 1 into the other equation and solve for the other variable
3. Substitute the value from step 2 into either original equation and solve to find the value of the variable is step 1.
Example 1:
Solve the following system
y=2x+1x-y=7
Exercise 1:
Solve the following system
y=x+2-x+y=4
Extended Response:
1. A gardener is planting two types of trees:
Type A is 36 inches tall and grows at a rate of 15 inches per year.
Type B is 48 inches tall and grows at a rate of 10 inches per year.
Algebraically determine exactly how many years it will take for these trees to be the same height.
Solving a System of Equations using Elimination
Directions
1. Look for opposites in either variable
2. Add to the equations vertically to eliminate one variable.
3. Solve the resulting equation for the other variable.
4. Substitute the value into either original equation to find the value of the eliminated variable.
Example 1
4x-2y=12x +2y=8
Exercise 1:
Solve each system by elimination
-5x+y= -35x-3y=-1
**Exercise 2:
4x+3y=112x-2y= -12
Extended Response
Julietta purchased a total of 10 pens and pencils for $4. Pens cost 50 cents and pencils cost 25 cents.
a) Write a system of equations to represent the situation.
b) Solve for how many pens and pencils Julietta purchased.
Challenge:
Isabella says that the two linear systems below have the same solution. Is she correct? Explain.
3x+2y=25x+4y=6 3x+2y=211x+8y=10