Word Gems: Mathematics: Notes: Equation, Variables on both sides

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Editor’s collection of notes:

Algebra

Equation, Variables on both sides

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from https://courses.lumenlearning.com/wm-prealgebra/chapter/solving-equations-with-variables-on-both-sides/

Our strategy will involve choosing one side of the equation to be the variable side, and the other side of the equation to be the constant side. This will help us with organization. Then, we will use the Subtraction and Addition Properties of Equality, step by step, to isolate the variable terms on one side of the equation.

Examples

Solve: $4 x + 6 = - 14$

Solution:

In this equation, the variable is only on the left side. It makes sense to call the left side the variable side. Therefore, the right side will be the constant side.

Since the left side is the variable side, the 6 is out of place. We must “undo” adding

6

by subtracting

6

, and to keep the equality we must subtract

6

from both sides. Use the Subtraction Property of Equality.

4 x + 6 - 6 = - 14 - 6


Simplify.		$4 x = - 20$


Now all the $x$

are on the left and the constant on the right.
Use the Division Property of Equality.		$\frac{4 x}{4} = \frac{- 20}{4}$


Simplify.		$x = - 5$


Check:		$4 x + 6 = - 14$


Let $x = - 5$

4 (- 5) + 6 = - 14


		$- 20 + 6 = - 14$


		$- 14 = - 14 ✓$

Solve: $2 y - 7 = 15$

Solution:
Notice that the variable is only on the left side of the equation, so this will be the variable side and the right side will be the constant side. Since the left side is the variable side, the $7$

is out of place. It is subtracted from the

2 y

, so to “undo” subtraction, add

7

to both sides.

2 y - 7

is the side containing a variable.

$15$

is the side containing only a constant.
Add $7$

to both sides.

2 y - 7 + 7 = 15 + 7


Simplify.		$2 y = 22$


The variables are now on one side and the constants on the other.
Divide both sides by $2$

\frac{2 y}{2} = \frac{22}{2}


Simplify.		$y = 11$


Check:		$2 y - 7 = 15$


Let $y = 11$

2 \cdot 11 - 7 \overset{?}{=} 15


		$22 - 7 \overset{?}{=} 15$


		$15 = 15 ✓$

Solve Equations with Variables on Both Sides

You may have noticed that in all the equations we have solved so far, we had variables on only one side of the equation. This does not happen all the time—so now we’ll see how to solve equations where there are variable terms on both sides of the equation. We will start like we did above—choosing a variable side and a constant side, and then use the Subtraction and Addition Properties of Equality to collect all variables on one side and all constants on the other side. Remember, what you do to the left side of the equation, you must do to the right side as well.

In the next example, the variable, $x$ , is on both sides, but the constants appear only on the right side, so we’ll make the right side the “constant” side. Then the left side will be the “variable” side.

Example

Solve: $5 x = 4 x + 7$

Solution:

5 x

is the side containing only a variable.

$4 x + 7$

is the side containing a constant.
We don’t want any variables on the right, so subtract the $4 x$

5 x - 4 x = 4 x - 4 x + 7


Simplify.		$x = 7$


We have all the variables on one side and the constants on the other. We have solved the equation.
Check:		$5 x = 4 x + 7$

Substitute

7

for

x

5 (7) \overset{?}{=} 4 (7) + 7


		$35 \overset{?}{=} 28 + 7$


		$35 = 35 ✓$

Solve: $7 x = - x + 24$

Solution:
The only constant, $24$

, is on the right, so let the left side be the variable side.

7 x

is the side containing only a variable.

$- x + 24$

is the side containing a constant.
Remove the $- x$

from the right side by adding

x

to both sides.

7 x + x = - x + x + 24


Simplify.	$8 x = 24$

All the variables are on the left and the constants are on the right. Divide both sides by

8

.	$\frac{8 x}{8} = \frac{24}{8}$


Simplify.	$x = 3$


Check:	$7 x = - x + 24$

Substitute

x = 3

.	$7 (3) \overset{?}{=} - (3) + 24$


	$21 = 21 ✓$

Did you see the subtle difference between the two equations? In the first, the right side looked like this: $4 x + 7$ , and in the second, the right side looked like this: $- x + 24$ , even though they look different, we still used the same techniques to solve both.

In the next example, you will see that it is beneficial to move the variable term to the right side of the equation. There is no “correct” side to move the variable term, but the choice can help you avoid working with negative signs.

Example

Solve: $5 y - 8 = 7 y$

Solution:
The only constant, $- 8$

, is on the left side of the equation, and the variable,

y

, is on both sides. Let’s leave the constant on the left and collect the variables to the right.

5 y - 8

is the side containing a constant.

$7 y$

is the side containing only a variable.
Subtract $5 y$

from both sides.

5 y - 5 y - 8 = 7 y - 5 y


Simplify.	$- 8 = 2 y$

We have the variables on the right and the constants on the left. Divide both sides by

2

.	$\frac{- 8}{2} = \frac{2 y}{2}$


Simplify.	$- 4 = y$


Rewrite with the variable on the left.	$y = - 4$


Check:	$5 y - 8 = 7 y$

Let

y = - 4

.	$5 (- 4) - 8 \overset{?}{=} 7 (- 4)$


	$- 20 - 8 \overset{?}{=} - 28$


	$- 28 = - 28 ✓$

Solve Equations with Variables and Constants on Both Sides

The next example will be the first to have variables and constants on both sides of the equation. As we did before, we’ll collect the variable terms to one side and the constants to the other side. You will see that as the number of variable and constant terms increases, so do the number of steps it takes to solve the equation.

Examples

Solve: $7 x + 5 = 6 x + 2$

Start by choosing which side will be the variable side and which side will be the constant side. The variable terms are $7 x$ and $6 x$ . Since $7$ is greater than $6$ , make the left side the variable side and so the right side will be the constant side.

7 x + 5 = 6 x + 2

Collect the variable terms to the left side by subtracting

6 x

from both sides.

7 x - 6 x + 5 = 6 x - 6 x + 2


Simplify.	$x + 5 = 2$

Now, collect the constants to the right side by subtracting

5

from both sides.

x + 5 - 5 = 2 - 5


Simplify.	$x = - 3$

The solution is

x = - 3

.
Check:	$7 x + 5 = 6 x + 2$

Let

x = - 3

.	$7 (- 3) + 5 \overset{?}{=} 6 (- 3) + 2$


	$- 21 + 5 \overset{?}{=} - 18 + 2$


	$16 = 16 ✓$

Solve: $6 n - 2 = - 3 n + 7$

We have

6 n

on the left and

- 3 n

on the right. Since

6 > - 3

, make the left side the “variable” side.

6 n - 2 = - 3 n + 7

We don’t want variables on the right side—add

3 n

to both sides to leave only constants on the right.

6 n + 3 n - 2 = - 3 n + 3 n + 7


Combine like terms.	$9 n - 2 = 7$

We don’t want any constants on the left side, so add

2

to both sides.

9 n - 2 + 2 = 7 + 2


Simplify.	$9 n = 9$

The variable term is on the left and the constant term is on the right. To get the coefficient of

n

to be one, divide both sides by

9

.	$\frac{9 n}{9} = \frac{9}{9}$


Simplify.	$n = 1$


Check:	$6 n - 2 = - 3 n + 7$

Substitute

1

for

n

.	$6 (1) - 2 \overset{?}{=} - 3 (1) + 7$


	$4 = 4 ✓$