Introduction to Systems of Linear Equations

Example 1: Are these two lines parallel, identical, or intersect at one point only? Does this system have a solution?

Y=2x+3

Y=2x-4

Solution: Since they have the same slope, but different y-intercepts, it follows that they are parallel. So, there is no solution.

Example 2: Are these two lines parallel, identical, or intersect at one point only? Does this system have a solution?

2x-3y=1

4x-6y=3

Solution: These two lines have the same slope. The slope of the first one: m=-2/-3=2/3. The slope of the second equation is m=-4/-6=2/3. But these lines are not the same because the second equation is not a multiple of the first equation. This means that once again, this system has no solution.

Example 3: Are these two lines parallel, identical, or intersect at one point only? Does this system have a solution?

x+y=1

-2x-2y=-2

Solution: These two equations are identical because if we divide both sides of the second equation by -2, we will get x+y=1, the first equation. Or you can take the first equation and multiply both sides by -2, getting the second equation. This means that there are infinitely many solutions.

Example 4: Are these two lines parallel, identical, or intersect at one point only? Does this system have a solution? If it has one, find it!

x-y=2

y=-x

Solution: These two lines intersect at one point only. We know this because they have different slopes. The first one has slope m=-1/-1=1, while the slope of the second equation m=-1 since the second equation is in slope-intercept form. So, let’s find the point of intersection!

We have three methods of solving:

(1) Graph, and note the point of intersection.

(2) Substitution

(3)Addition/Elimination

Using (1), the graphing technique, we note that the point of intersection is (1,-1)

Using method (2) to find this solution, we will substitute the second equation into the first one:

First Equation: x-y=2

Second Equation y=-x

x-(-x)=2 --> 2x=2  x=1

Now that we know that the x-coordinate of the solution is x=1, to find the y-coordinate, plug in the value x=1 into any one of the equations:

Y=(-1)

So, the solution is (1,-1)

Example 5: Solve:

x+y=-4

x=-3y

Use two methods: (1) Graphing, and (2) Substitution

So, the solution is (-6,2)

Let’s now use the method of substitution:

-3y+y=-4  -2y=-4  y=2  x=-3(2)=-6

Example 6: Solve the following system using all three methods:

2x-3y=1

-2x+y=5

Method 1: Graphing

Graphing First Line: We know that the slope is m=2/3. We now have to find a point on this line to graph it. We will not be looking for the x and y intercepts because the coefficients of x, and y are not factors of the constant 1. Instead, let x=1, we get 2(1)-3y=1 -3y=-1,  y=1/3. This didn’t work! Let’s try x=2: 2(2)-3y=1  4-3y=1  -3y=1-4  -3y=-3  y=1. So, the point (2,1) is on the line.

Graphing Second Line: We can find quickly the y-intercept

By letting x=0, we can find y: -2(0)+y=5  y=5 So, the point (0,5) is on this line. We also know the slope of the line, m=-(-2)/1=2. So, we will use the point (0,5) and the slope 2 to graph the second line.

So, the solution is (-4,-3)

Method 2: Substitution

2x-3y=1

-2x+y=5

To use substitution, we have to pick an equation, and isolate one of the variables. Isolating the variable y in the second equation gives y=2x+5. We substitute this expression (2x+5) into the variable y in the first equation:

2x-3(2x+5)=1  2x-6x-15=1 -4x=16  x=-4

To find y, we will substitute x=-4 into -2(-4)+y=5  8+y=5  y=-3

SO, the solution is (-4,-3)

Method 3: Addition/Elimination: We notice that the two coefficients for x are opposites. So, we will add the two equations:

2x-3y=1

-2x+y=5

- 2y =6

-2y=6 y=-3.

Now, we substitute y=-3 into the second equation, to get: -2x-3=5 -2x=8  x=-4

Answer: (-4,-3)