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Solving Linear Equations with Variables on Both Sides: A Step-by-Step Guide

Solving linear equations with variables on both sides is a fundamental skill in algebra that I consider essential for understanding a wide range of mathematical and real-world problems.

An equation of this sort typically looks like $ax + b = cx + d$, where $x$ i s the variable you’re looking to solve for, and $a$, $b$, $c$ , and $d$ represent constants.

My general strategy for tackling these equations involves a series of steps to isolate the variable on one side of the equation.

This means I’ll be adding, subtracting, multiplying, or dividing throughout to get $x$ by itself, which simplifies the equation down to something like $x = e$ , with $e$ being the solution.

In dealing with these types of equations, it’s important to maintain the balance of the equation , as whatever I do to one side, I must also do to the other to preserve equality.

Sometimes, this involves combining like terms or using the distributive property to simplify each side of the equation ; it’s like a mathematical dance where precision and balance are key.

Remember to check your solution by substituting it back into the original equation to ensure both sides equal out. Stay tuned as I walk you through this interesting and engaging process, sparking a light of understanding in the elegant dance of algebra.

Steps for Solving Equations With Variables on Both Sides

When I encounter an algebraic equation with variables on both sides, my first step is to simplify each side of the equation if needed. This involves expanding expressions using the distributive property , combining like terms , and organizing the equation so it becomes easier to work with.

Here’s a breakdown in a table format:

Let’s consider the equation, $7y = 11x + 4x – 8$. I would combine the $x$ terms on the right first, getting $7y = 15x – 8$. Now, if I need $y$ on one side and $x$ on the other, I might decide to move the $x$ terms to the other side by subtracting $15x$ from both sides, resulting in $7y – 15x = -8$.

After that, if there are any coefficients attached to the variable I’m solving for, I’ll divide both sides by that coefficient. If I am solving for $y$, and the equation is $7y – 15x = -8$, I’d divide everything by 7 to isolate $y$:

$$ \frac{7y}{7} – \frac{15x}{7} = \frac{-8}{7} $$ $$ y – \frac{15x}{7} = \frac{-8}{7} $$

Now, if I need to solve for $y$ explicitly, I might rearrange the terms to show $y$ as a function of $x$, which would give me $y = \frac{15x}{7} – \frac{8}{7}$ as the final step.

This straightforward approach ensures the equation is balanced and leaves me with a clear solution for the variable in question.

Solving Example Equation

When I solve equations with variables on both sides , I start by simplifying both sides separately. Let’s take the example of the linear equation $2x + 3 = x – 5$. My goal is to isolate the variable , x, on one side to find its value.

First, I use addition and subtraction properties of equality to get all the x terms on one side and the constants on the other. Subtracting x from both sides gives me $2x – x + 3 = x – x – 5$, which simplifies to $x + 3 = -5$.

Next, I apply subtraction to remove 3 from both sides, resulting in $x + 3 – 3 = -5 – 3$. Simplifying this, I get $x = -8$.

Here’s a breakdown of the steps I took in a more visual format:

After simplifying, I realize that the equation has led me to a true statement : $x = -8$ is the solution indicating the value of the variable that satisfies the equation .

It’s also important to combine like terms and utilize the distributive property appropriately if the equation were to have more complex expressions.

By performing the correct multiplication or division based on the properties of equality , I ensure the equation remains balanced and arrive at an accurate solution.

Solving linear equations with variables on both sides has been a focus throughout this discussion.

I’ve demonstrated how to bring the variables to one side and the constants to the other, which often involves the use of the addition or subtraction property of equality. Remember to always perform the same operation on both sides to maintain the balance of the equation.

Once the variables are isolated, you might need to use the division property to find the variable’s value. The key to solving these equations lies in following a systematic approach: distribute, combine like terms, and isolate the variable .

For example, when presented with an equation like $3x + 7 = 2x – 5$ , my first step is to eliminate the $x$ from one side by subtracting $2x$ from both sides, simplifying it to $x + 7 = -5$ . Then, I isolate $x$ by subtracting $7$ from both sides, resulting in $x = -12$ .

By taking these steps, I ensure that each equation is solved accurately, and the true value of the variable is revealed.

Finally, checking my work by substituting the solution back into the original equation guarantees the integrity of the results.

This methodical approach not only makes complex problems more manageable but also instills confidence in my ability to tackle similar challenges in the future.

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Module 6: Growth Models

Solving equations with variables on both sides, learning outcomes.

• Identify the constant and variable terms in an equation
• Solve linear equations by isolating constants and variables
• Solve linear equations with variables on both sides that require several steps

The equations we solved in the last section simplified nicely so that we could use the division property to isolate the variable and solve the equation. Sometimes, after you simplify you may have a variable and a constant term on the same side of the equal sign.

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.

Read on to find out how to solve this kind of equation.

Solve: [latex]4x+6=-14[/latex]

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.

Solve: [latex]2y - 7=15[/latex]

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 [latex]7[/latex] is out of place. It is subtracted from the [latex]2y[/latex], so to “undo” subtraction, add [latex]7[/latex] to both sides.

Now you can try a similar problem.

Try It

https://ohm.lumenlearning.com/multiembedq.php?id=142131&theme=oea&iframe_resize_id=mom2

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, [latex]x[/latex], 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.

Solve: [latex]5x=4x+7[/latex]

Solve: [latex]7x=-x+24[/latex].

Solution: The only constant, [latex]24[/latex], is on the right, so let the left side be the variable side.

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

Now you can try solving an equation with variables on both sides where it is beneficial to move the variable term to the left side.

try it

https://ohm.lumenlearning.com/multiembedq.php?id=142129&theme=oea&iframe_resize_id=mom3

https://ohm.lumenlearning.com/multiembedq.php?id=142132&theme=oea&iframe_resize_id=mom4

In our last examples, we moved the variable term to the left side of the equation. 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.

Solve: [latex]5y - 8=7y[/latex]

Solution: The only constant, [latex]-8[/latex], is on the left side of the equation, and the variable, [latex]y[/latex], is on both sides. Let’s leave the constant on the left and collect the variables to the right.

Now you can try solving an equation where it is beneficial to move the variable term to the right side.

https://ohm.lumenlearning.com/multiembedq.php?id=142125&theme=oea&iframe_resize_id=mom2

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.

Solve: [latex]7x+5=6x+2[/latex]

Solution: Start by choosing which side will be the variable side and which side will be the constant side. The variable terms are [latex]7x[/latex] and [latex]6x[/latex]. Since [latex]7[/latex] is greater than [latex]6[/latex], make the left side the variable side and so the right side will be the constant side.

Solve: [latex]6n - 2=-3n+7[/latex]

In the following video we show an example of how to solve a multi-step equation by moving the variable terms to one side and the constants to the other side. You will see that it doesn’t matter which side you choose to be the variable side; you can get the correct answer either way.

In the next example, we move the variable terms to the right side to keep a positive coefficient on the variable.

Solve: [latex]2a - 7=5a+8[/latex]

This equation has [latex]2a[/latex] on the left and [latex]5a[/latex] on the right. Since [latex]5>2[/latex], make the right side the variable side and the left side the constant side.

The following video shows another example of solving a multi-step equation by moving the variable terms to one side and the constants to the other side.

Try these problems to see how well you understand how to solve linear equations with variables and constants on both sides of the equal sign.

https://ohm.lumenlearning.com/multiembedq.php?id=142134&theme=oea&iframe_resize_id=mom20

https://ohm.lumenlearning.com/multiembedq.php?id=142136&theme=oea&iframe_resize_id=mom200

We just showed a lot of examples of different kinds of linear equations you may encounter. There are some good habits to develop that will help you solve all kinds of linear equations. We’ll summarize the steps we took so you can easily refer to them.

Solve an equation with variables and constants on both sides

• Choose one side to be the variable side and then the other will be the constant side.
• Collect the variable terms to the variable side, using the Addition or Subtraction Property of Equality.
• Collect the constants to the other side, using the Addition or Subtraction Property of Equality.
• Make the coefficient of the variable [latex]1[/latex], using the Multiplication or Division Property of Equality.
• Check the solution by substituting it into the original equation.
• Question ID 142131, 142125, 142129, 142132, 142134, 142136. Authored by : Lumen Learning. License : CC BY: Attribution . License Terms : IMathAS Community License, CC-BY + GPL
• Solve a Linear Equation in One Variable with Variables on Both Sides: 2m-9=6m-17. Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/kiYPW6hrTS4 . License : CC BY: Attribution
• Solve a Linear Equation in One Variable with Variables on Both Sides: 2x+8=-2x-24. Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/_hBoWoctfAo . License : CC BY: Attribution
• Ex: Solve an Equation with Variable Terms on Both Sides. Authored by : James Sousa (Mathispower4u.com). Located at : https://youtu.be/xfXGgqgJyDE . License : CC BY: Attribution
• Prealgebra. Provided by : OpenStax. License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]

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9.3: Solve Equations with Variables and Constants on Both Sides

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Learning Objectives

By the end of this section, you will be able to:

• Solve an equation with constants on both sides
• Solve an equation with variables on both sides
• Solve an equation with variables and constants on both sides

Before you get started, take this readiness quiz.

• Simplify: 4y−9+9. If you missed this problem, review Exercise 1.10.20 .

Solve Equations with Constants on Both Sides

In all the equations we have solved so far, all the variable terms were on only one side of the equation with the constants on the other side. This does not happen all the time—so now we will learn to solve equations in which the variable terms, or constant terms, or both are on both sides of the equation.

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.” Then, we will use the Subtraction and Addition Properties of Equality to get all the variable terms together on one side of the equation and the constant terms together on the other side.

By doing this, we will transform the equation that began with variables and constants on both sides into the form \(ax=b\). We already know how to solve equations of this form by using the Division or Multiplication Properties of Equality.

Example \(\PageIndex{1}\)

Solve: \(7x+8=−13\).

In this equation, the variable is found 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. We will write the labels above the equation to help us remember what goes where.

Since the left side is the “xx”, or variable side, the 8 is out of place. We must “undo” adding 8 by subtracting 8, and to keep the equality we must subtract 8 from both sides.

Try It \(\PageIndex{2}\)

Solve: \(3x+4=−8\).

\(x=−4\)

Try It \(\PageIndex{3}\)

Solve: \(5a+3=−37\).

\(a=−8\)

Example \(\PageIndex{4}\)

Solve: \(8y−9=31\).

Notice, the variable is only on the left side of the equation, so we will call this side the “variable” side, and the right side will be the “constant” side. Since the left side is the “variable” side, the 9 is out of place. It is subtracted from the 8y, so to “undo” subtraction, add 9 to both sides. Remember, whatever you do to the left, you must do to the right.

Try It \(\PageIndex{5}\)

Solve: \(5y−9=16\).

Try It \(\PageIndex{6}\)

Solve: \(3m−8=19\).

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. It may take several steps to solve this equation, so we need a clear and organized strategy.

Example \(\PageIndex{7}\)

Solve: \(9x=8x−6\).

Here the variable is on both sides, but the constants only appear on the right side, so let’s make the right side the “constant” side. Then the left side will be the “variable” side.

Try It \(\PageIndex{8}\)

Solve: \(6n=5n−10\).

\(n = -10\)

Try It \(\PageIndex{9}\)

Solve: \(-6c = -7c - 1\)

Example \(\PageIndex{10}\)

Solve: \(5y - 9 = 8y\)

The only constant is on the left and the y’s are on both sides. Let’s leave the constant on the left and get the variables to the right.

Try It \(\PageIndex{11}\)

Solve: \(3p−14=5p\).

Try It \(\PageIndex{12}\)

Solve: \(8m + 9 = 5m\)

Example \(\PageIndex{13}\)

Solve: \(12x = -x + 26\)

The only constant is on the right, so let the left side be the “variable” side.

Try It \(\PageIndex{14}\)

Solve: \(12j = -4j + 32\)

Try It \(\PageIndex{15}\)

Solve: \(8h = -4h + 12\)

Example \(\PageIndex{16}\): How to Solve Equations with Variables and Constants on Both Sides

Solve: \(7x + 5 = 6x + 2\)

Try It \(\PageIndex{17}\)

Solve: \(12x+8=6x+2\).

\(x=−1\)

Try It \(\PageIndex{18}\)

Solve: \(9y+4=7y+12\).

​​​​​​ We’ll list the steps below so you can easily refer to them. But we’ll call this the ‘Beginning Strategy’ because we’ll be adding some steps later in this chapter.

BEGINNING STRATEGY FOR SOLVING EQUATIONS WITH VARIABLES AND CONSTANTS ON BOTH SIDES OF THE EQUATION.

• Choose which side will be the “variable” side—the other side will be the “constant” side.
• Collect the variable terms to the “variable” side of the equation, using the Addition or Subtraction Property of Equality.
• Collect all the constants to the other side of the equation, using the Addition or Subtraction Property of Equality.
• Make the coefficient of the variable equal 1, using the Multiplication or Division Property of Equality.
• Check the solution by substituting it into the original equation.

In Step 1, a helpful approach is to make the “variable” side the side that has the variable with the larger coefficient. This usually makes the arithmetic easier.

Example \(\PageIndex{19}\)

Solve: \(8n−4=−2n+6\).

In the first step, choose the variable side by comparing the coefficients of the variables on each side.

Try It \(\PageIndex{20}\)

Solve: \(8q - 5 = -4q + 7\)

Try It \(\PageIndex{21}\)

Solve: \(7n - 3 = n + 3\)

Example \(\PageIndex{22}\)

Solve: \(7a -3 = 13a + 7\)

Since 13>7, make the right side the “variable” side and the left side the “constant” side.

Try It \(\PageIndex{23}\)

Solve: \(2a - 2 = 6a + 18\)

Try It \(\PageIndex{24}\)

Solve: \(4k -1 = 7k + 17\)

In the last example, we could have made the left side the “variable” side, but it would have led to a negative coefficient on the variable term. (Try it!) While we could work with the negative, there is less chance of errors when working with positives. The strategy outlined above helps avoid the negatives!

To solve an equation with fractions, we just follow the steps of our strategy to get the solution!

Example \(\PageIndex{25}\)

Solve: \(\frac{4}{5}x + 6 = \frac{1}{4}x - 2\)

Since \(\frac{5}{4} > \frac{1}{4}\), make the left side the “variable” side and the right side the “constant” side.

Try It \(\PageIndex{26}\)

Solve: \(\frac{7}{8}x - 12 = -\frac{1}{8}x - 2\)

Try It \(\PageIndex{27}\)

Solve: \(\frac{7}{6}x + 11 = \frac{1}{6}y + 8\)

We will use the same strategy to find the solution for an equation with decimals.

Example \(\PageIndex{28}\)

Solve: \(7.8x+4=5.4x−8\).

Since \(7.8>5.4\), make the left side the “variable” side and the right side the “constant” side.

Try It \(\PageIndex{29}\)

Solve: \(2.8x + 12 = -1.4x - 9\)

Try It \(\PageIndex{30}\)

Solve: \(3.6y + 8 = 1.2y - 4\)

Key Concepts