Writing and Solving Linear Equations
In the previous investigations, if we knew the value of one variable, we could use a table or a graph to find the
value of the other variable. For example, if Alana raised $8 from a sponsor, then how many kilometers did she
walk?
We can check our solution, (6, 8) in the equation y = 5 + 0.5x. But, what if we didn’t want to make the table
and graph first to answer our original question. Can we find the distance by solving the equation instead of
using a table or graph?.
What if Alana raised $10 from her sponsor? How far did she walk?
10 = 5 + 0.5x
Check your solution on the graph.
Solving Two-Step Equations
A two-step equation is as straightforward as it sounds. You will need to perform two steps in order to solve the
equation.
One goal in solving an equation is to have only variable on one side of the equal sign and numbers on the other
side of the equal sign. The other goal is to have the number in front of the variable equal to one. The variable
does not always have to be x. These equations can make use of any letter as a variable.
The strategy for getting the variable by itself with a coefficient of 1 involves using opposite operations. For
example, to move something that is added to the other side of the equation, you should subtract. The most
important thing to remember in solving a linear equation is that whatever you do to one side of the equation,
you MUST do to the other side. So if you subtract a number from one side, you MUST subtract the same value
from the other side. You will see how this works in the example.
Let's Practice:
This problem does not have the variable by itself on one side. We need to get rid of the 4 that is added, so we’ll
need to subtract 4 from both sides. Even after doing that, there is still a 3 multiplied by the variable, so division
will be necessary to eliminate it.
Solve 3x + 4 = 19
Solve the equations below. Support your answers.
a) 5x – 6 = 29
b)
𝒄
𝟑
- 13 = 0
c) 15 = 6d - 9
In the previous investigations, we identified the point of intersection as the value where two relationships are
equal. For example, in Carly and Courtney’s walking race, at the point (30, 75), the two girls were tied with
each other. You needed to know this point of intersection in order to identify the race’s distance so Courtney
could win. Most of you spent a length amount of time using tables, graphs, and the guess/check method to find
this distance. Let’s look at how solving an equation is the easiest method:
2.5x = 45 + x
Equations with x's on BOTH sides of the equal sign:
You need to "Get the x's on one side and the numbers on the other." Then you can solve.
Example:
12x – 11 = 7x + 9
-7x
-7x
5x – 11 = 9
+11 +11
5x
= 20
5
5
x=4
Move the x’s to one side.
Now it looks like a multistep equation that we did in the 1st section.
Add to get rid of the subtraction.
Now divide to get rid of the multiplication
Solve the Multi-Step Equations and check:
1) 15x - 24 = -79 + 4x
2)
11x - 3 = 7x + 25