GED Mathematical Reasoning: Systems Of Linear Equations

A system of linear equations refers to a set of two or more linear equations. Solving a system of linear equations means to find the ordered pair that satisfies each equation in the system. In other words, the solution to a system of linear equations is the ordered pair that makes each equation a true statement.

 

Solving a system of two linear equations using substitution

  • Step 1: Solve one of the equations for one of the variables.
  • Step 2:  Substitute the expression from Step 1 into the other equation.
  • Step 3:  Solve the equation from Step 2.
  • Step 4:  Substitute the value found in Step 3 into either equation to find the other coordinate of the solution.
  • Step 5:  Check the solution. Plug the x- and y-value of the solution into each equation and verify that they make each equation a true statement.

 

Example 1

Solve the system of equations by graphing:

 y = -2x + 4
 8x + 2y = 20

 

In example 1, we’re asked to solve the system of equations by graphing. This is actually a pretty simple process. We’ll graph each equation using a table of values and then look for where the lines intersect. The ordered pair representing the intersection point is the solution to the system of equations.

We’ll start by graphing the equation y = -2x + 4.

A good first step when graphing any equation using a table of values is to solve the equation for “y”, which is already done in this first equation. For our table of values let’s choose x-values:  -1, zero and 1. One at a time, we’ll substitute these x-values into the equation and simplify to find the corresponding y-value. We’ll then use the ordered pair points to sketch the graph.

When we let x = -1, we will simplify the right side of the equal sign to determine the corresponding y value. y = -2(-1) + 4 = 2 + 4 = 6. The ordered pair point is: (-1,6).

Next we’ll let x = 0. y = -2(0) + 4 = 0 + 4 = 4. The ordered pair point is: (0, 4).

Now we’ll let x = 1. y = -2(1) + 4 = -2 + 4 = 2. So the ordered pair point is: (1, 2).

x Substitute x and solve for y y Ordered Pair (Point)
-1 y = -2(-1) + 4 = 2 + 4 = 6 6 (-1, 6)
0 y = -2(0) + 4 = 0 + 4 = 4 4 (0, 4)
1 y = -2(1) + 4 = -2 + 4 = 2 2 (1, 2)

Plotting these three points results in the line shown.

coord20

Next let’s graph the second equation: 8x + 2y = 20. Although it isn’t necessary, let’s solve this equation for y. In most cases, this makes completing the table a bit easier.

8x + 2y = 20

2y = -8x + 20

y = -4x + 10

For our table of values let’s choose x-values:  2, 3 and 4

When we let x = 2, y = -4(2) + 10 = -8 + 10 = 2. And the ordered pair point is: (2, 2)

Next we’ll let x = 3. y = -4(3) + 10 = -12 + 10 = -2. The ordered pair point is: (3, -2)

Now we’ll let x = 4. y = -4(4) + 10 = -16 + 10 = -6. So the ordered pair point is: (4, -6)

x Substitute x and solve for y y Ordered Pair (Point)
2 y = -4(2) + 10 = -8 + 10 = 2 2 (2, 2)
3 y = -4(3) + 10 = -12 + 10 = -2 -2 (3, -2)
4 y = -4(4) + 10 = -16 + 10 = -6 -6 (4, -6)

Let’s plot this line on the same graph as the first line so that we can see where the two lines intersect. I’ll show this line in red.

coord21

The two lines intersect at the point (3, -2), which means the solution to this system of two linear equations is the ordered pair: (3, -2).

coord22

Let’s check this ordered pair solution to ensure that it satisfies both equations. When we let x = 3 and y = -2 in each equation separately, we should get a true statement in both cases.

When we substitute the ordered pair (3, -2) in the first equation, we get:

Check: y = -2x + 4
-2 = -2(3) + 4
-2 = -6 + 4
-2 = -2 which is true

Since -2 equals -2 is a true statement, the ordered pair satisfies this first equation.

Now let’s check the solution in the second equation. When we substitute the ordered pair (3, -2) in the second equation, we get:

Check: 8x + 2y = 20
8(3) + 2 (-2) = 20
24 - 4 = 20
20 = 20 which is true

Since 20 equals 20 is a true statement, the ordered pair satisfies this second equation as well and we can be confident that our solution is correct.

 

Example 2

Solve the system of equations from Example 1 using substitution.

 y = -2x + 4
 8x + 2y = 20

 

Solving a system of two linear equations using substitution is a more algebraic approach. And it is a good method to know because it is more accurate – especially if one or both coordinates in the ordered pair solution is not an integer.

The first step to solving a system of linear equations using the substitution method is to solve one of the equations for one of the variables. In this example, the first equation is already solved for y so this step is already done for us.

The second step is to substitute the expression from step one into the other equation. In other words, since the first equation tells us that y is equal to the expression: -2x + 4, we will substitute -2x + 4 for y in the second equation.

coord23This gives us the equation:

8x + 2(-2x + 4) = 20

Now we have an equation that contains only the variable x, which means we can solve it for x. In fact, the third step of the process is to solve the equation from Step 2.

8x - 4x + 8 = 20

4x + 8 = 20

4x = 12

x = 3

Now that we have the x-coordinate of our ordered pair solution, we need to calculate the y-coordinate. This brings us to Step 4, which is to substitute the value found in step 3 into either equation to find the other coordinate of the solution. We can substitute “x” equals 3 back into either equation, but using the first equation will be more efficient.

If we let x = 3 in the first equation, we get:

y = -2(3) + 4

y = -6 + 4

y = -2

So the ordered pair solution is: (3, -2)

The final step is to substitute these values for “x” and “y” back into each equation separately to check that they make each equation a true statement.

We already checked this solution in Example 1.

 

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