Solving and Justifying Multi-Step Equations | Grade 8 Math | LevelKite

An equation says that two expressions have the same value, for example $2x + 3 = 11$ . In this lesson each variable, such as $x$ , stands for an unknown number, and it may be any real number (any number on the number line). These allowed numbers are the permitted values of the variable.

A solution is a permitted value that makes the equation true. For $2x + 3 = 11$ , the number $4$ is a solution because $2 \cdot 4 + 3 = 11$ .

A multi-step equation needs several moves, not just one or two, before the variable stands alone. You solve it by changing it, one move at a time, into simpler equations that have exactly the same solutions. Equations with exactly the same solutions are called equivalent equations. For example, $2x + 3 = 11$ , $2x = 8$ and $x = 4$ are equivalent, because $4$ is the only solution of each one.

Key concept

Legal moves keep the solutions the same

A legal move is a change to an equation that gives an equivalent equation. There are two kinds.

Rewrite one side as an equivalent expression. Equivalent expressions have the same value for every permitted value of the variable, so the side keeps its value.
Do the same operation to both sides: add or subtract the same number or term, or multiply or divide by the same number that is not zero.

Equal values stay equal when you treat them the same way, and each move can be undone by its opposite move. So no solution is lost and no new one appears. Zero is not allowed because you cannot divide by zero, and multiplying by zero cannot be undone.

Here is a plan for multi-step equations like the ones in this lesson.

Simplify each side. Use the distributive property to remove parentheses. Then combine like terms. Terms are the parts of an expression that are added or subtracted. Like terms have the same variable part, such as $6x$ and $-4x$ , or are both plain numbers, such as $-10$ and $4$ .
Collect. Add or subtract on both sides until the variable terms are on one side and the plain numbers are on the other.
Isolate. Divide both sides by the coefficient of the variable, which is the number that multiplies it. You may do this only when the coefficient is not zero.
Check. Replace the variable with your value in the original equation. It is a solution only if both sides give the same number. Use the original equation because a later line could already contain a mistake.

Steps 1 to 3 use only legal moves, so each new line has the same solutions as the line before it.

Worked example

Solve a multi-step equation

Solve $2(3x - 5) + 4 = 4x + 8$ , where $x$ may be any real number.

Simplify the left side with the distributive property: 2 times 3x is 6x, and 2 times -5 is -10. The new expression has the same value as the old one for every permitted value of x, so the solutions do not change.
$6x - 10 + 4 = 4x + 8$
Combine the like terms -10 and 4 on the left side. This also rewrites the left side as an equivalent expression.
$-10 + 4 = -6 \quad\text{so}\quad 6x - 6 = 4x + 8$
Collect the variable terms on the left by subtracting 4x from both sides. Both sides lose the same amount, so they are equal for exactly the same values of x as before.
$6x - 6 - 4x = 4x + 8 - 4x \quad\text{so}\quad 2x - 6 = 8$
Add 6 to both sides so that the variable term stands alone. Both sides gain the same amount, so the solutions stay the same.
$2x - 6 + 6 = 8 + 6 \quad\text{so}\quad 2x = 14$
Isolate x by dividing both sides by 2, the coefficient of x. This is a legal move because 2 is not zero.
$\frac{2x}{2} = \frac{14}{2} \quad\text{so}\quad x = 7$
Check in the original equation by replacing x with 7 on each side. Both sides equal 36, so 7 makes the equation true.
$2(3 \cdot 7 - 5) + 4 = 2(16) + 4 = 36 \quad\text{and}\quad 4 \cdot 7 + 8 = 28 + 8 = 36$

Answer

The solution is $x = 7$ , because it makes both sides of the original equation equal to $36$ .

A rational number is a number that can be written as a fraction of two integers with a denominator that is not zero. Integers, fractions such as $\frac{3}{4}$ and decimals such as $0.5$ and $-1.3$ are all rational. So $\frac{3}{4}x - 2 = \frac{1}{6}x + 5$ is an equation with rational coefficients.

The legal moves stay exactly the same. Only the arithmetic changes, and you can handle it in two ways.

Work with the fractions or decimals as they are, using the usual rules for adding, subtracting, multiplying and dividing them.
Clear the fractions first. This is an extra move before step 1 of the plan. Multiply both sides by the least common denominator (LCD), the smallest positive integer that every denominator divides evenly. The LCD is never zero, so this is a legal move. You must multiply every term on both sides. Each denominator divides the LCD, so every fraction becomes an integer.

Worked example

Clear fractions with the least common denominator

Solve $\frac{3}{4}x - 2 = \frac{1}{6}x + 5$ .

The denominators are 4 and 6. The smallest positive integer that both divide evenly is 12, so the LCD is 12. Multiply both sides by 12. Because 12 is not zero, the new equation has the same solutions.
$12\left(\tfrac{3}{4}x - 2\right) = 12\left(\tfrac{1}{6}x + 5\right)$
Use the distributive property so that 12 multiplies every term on both sides, including the whole numbers 2 and 5. Each side becomes an equivalent expression.
$12 \cdot \tfrac{3}{4}x - 12 \cdot 2 = 12 \cdot \tfrac{1}{6}x + 12 \cdot 5 \quad\text{so}\quad 9x - 24 = 2x + 60$
Subtract 2x from both sides to collect the variable terms on the left. The same operation on both sides keeps the solutions the same.
$9x - 24 - 2x = 2x + 60 - 2x \quad\text{so}\quad 7x - 24 = 60$
Add 24 to both sides, again the same operation on each side, so that the variable term stands alone.
$7x - 24 + 24 = 60 + 24 \quad\text{so}\quad 7x = 84$
Divide both sides by 7, which is a legal move because 7 is not zero.
$\frac{7x}{7} = \frac{84}{7} \quad\text{so}\quad x = 12$
Check in the original equation by replacing x with 12 on each side. Both sides equal 7, so 12 makes the equation true.
$\tfrac{3}{4}(12) - 2 = 9 - 2 = 7 \quad\text{and}\quad \tfrac{1}{6}(12) + 5 = 2 + 5 = 7$

Answer

The solution is $x = 12$ , because both sides of the original equation equal $7$ when $x = 12$ .

To justify a step is to name the rule that makes it a legal move. A justified solution is an argument: anyone can read down it and see that each line has the same solutions as the line before it.

Every reason is one of two kinds. Either you rewrote one side as an equivalent expression, or you did the same operation to both sides. A rule of the second kind is called a property of equality. You have already used all of these moves in the examples above. The table gives the name to write for each move and the condition it needs.

Reasons that justify a step

What you do	Reason to write	Condition for a legal move
Multiply out parentheses on one side	Distributive property	The new expression equals the old one for every permitted value
Add or subtract like terms on one side	Combine like terms	The new expression equals the old one for every permitted value
Add the same number or term to both sides	Addition property of equality	Use any number or term, such as $6$ or $2x$
Subtract the same number or term from both sides	Subtraction property of equality	Use any number or term, such as $6$ or $2x$
Multiply both sides by the same number	Multiplication property of equality	The number must not be $0$
Divide both sides by the same number	Division property of equality	The number must not be $0$

Worked example

Solve with decimals and justify every step

Solve $0.5(x - 4) = 0.2x + 1.3$ and give the reason for each step.

Distributive property. Multiply both x and -4 by 0.5 to rewrite the left side as an equivalent expression.
$0.5 \cdot x - 0.5 \cdot 4 = 0.2x + 1.3 \quad\text{so}\quad 0.5x - 2 = 0.2x + 1.3$
Subtraction property of equality. Subtract 0.2x from both sides to collect the variable terms on the left.
$0.5x - 2 - 0.2x = 0.2x + 1.3 - 0.2x \quad\text{so}\quad 0.3x - 2 = 1.3$
Addition property of equality. Add 2 to both sides.
$0.3x - 2 + 2 = 1.3 + 2 \quad\text{so}\quad 0.3x = 3.3$
Division property of equality. Divide both sides by 0.3, which is not zero. Dividing 3.3 by 0.3 gives the same result as dividing 33 by 3.
$\frac{0.3x}{0.3} = \frac{3.3}{0.3} \quad\text{so}\quad x = 11$
Check in the original equation by replacing x with 11 on each side. Both sides equal 3.5, so 11 makes the equation true.
$0.5(11 - 4) = 0.5(7) = 3.5 \quad\text{and}\quad 0.2(11) + 1.3 = 2.2 + 1.3 = 3.5$

Answer

The solution is $x = 11$ . Each line follows from the line before it by the distributive property or a property of equality.

When every line of a solution is equivalent to the line before it, the last line has the same solutions as the original equation. So a wrong answer means that at least one line is not equivalent to the line before it. The lines after it carry the mistake along, even when those later moves are legal.

To find and correct an error:

Check the answer in the original equation. If the two sides give different numbers, the value is not a solution, so some line contains an error.
Read down the work. For each line, name the legal move that should produce it from the line above, and carry out that move yourself. The first line that does not match your result is where the work breaks.
Correct that line, redo the lines after it, and check the new answer in the original equation.

Worked example

Find and correct an error

Maya solved $\frac{1}{2}(4x - 6) = 5x + 18$ in four lines: (1) $2x - 3 = 5x + 18$ , (2) $7x - 3 = 18$ , (3) $7x = 21$ , (4) $x = 3$ . Find her error, explain why it is an error, and solve the equation correctly.

Check Maya's answer in the original equation by replacing x with 3 on each side. The left side is 3 and the right side is 33. They are different, so 3 is not a solution and at least one line contains an error.
$\tfrac{1}{2}(4 \cdot 3 - 6) = \tfrac{1}{2}(6) = 3 \quad\text{but}\quad 5 \cdot 3 + 18 = 15 + 18 = 33$
Read down the work. Line 1 is justified by the distributive property: half of 4x is 2x and half of -6 is -3. Line 2 is not justified. The 5x is gone from the right side, so 5x was subtracted there. On the left, 2x became 7x, so 5x was added there. The two sides did not get the same operation, so no property of equality produces line 2. That is the error.
$\text{left: } 2x + 5x = 7x \qquad \text{right: } 5x - 5x = 0$
Correct line 2. The legal move is the subtraction property of equality: subtract 5x from both sides.
$2x - 3 - 5x = 5x + 18 - 5x \quad\text{so}\quad -3x - 3 = 18$
Addition property of equality. Add 3 to both sides.
$-3x - 3 + 3 = 18 + 3 \quad\text{so}\quad -3x = 21$
Division property of equality. Divide both sides by -3. Dividing by a negative number is allowed because it is not zero. A positive number divided by a negative number is negative.
$\frac{-3x}{-3} = \frac{21}{-3} \quad\text{so}\quad x = -7$
Check the new answer in the original equation by replacing x with -7 on each side. Both sides equal -17, so -7 makes the equation true.
$\tfrac{1}{2}(4(-7) - 6) = \tfrac{1}{2}(-34) = -17 \quad\text{and}\quad 5(-7) + 18 = -35 + 18 = -17$

Answer

Maya's error is in line 2: she added $5x$ on the left side but subtracted $5x$ on the right side. Subtracting $5x$ from both sides gives the correct solution, $x = -7$ .

Takeaway. Solving a multi-step equation means writing a chain of equivalent equations. Each link is one of two legal moves: rewriting a side as an equivalent expression, or doing the same operation to both sides without multiplying or dividing by zero. Rational coefficients change the arithmetic, not the moves. Naming the reason for each line justifies your work, and it shows you the exact line where a wrong solution breaks. Always finish by checking your answer in the original equation.

Next, you will use these moves to solve word problems that lead to multi-step equations.