Inverse Operations and Error Checks | Grade 8 Math | LevelKite

You already know the properties of equality: doing the same valid operation to both sides of an equation keeps the two sides equal, and dividing by zero is never valid. Now you will use that idea to solve equations. To solve an equation is to find every value of the variable that makes it true. Each of those values is a solution.

Take $x + 8 = 3$ . The variable $x$ stands for an unknown number. In this lesson a variable may be any real number, which means any number on the number line, including negative numbers, fractions and decimals. These are its permitted values. The equation asks a question: which number gives $3$ after you add $8$ to it?

To answer, undo what was done to $x$ . Inverse operations are operations that undo each other: adding and subtracting the same number, or multiplying and dividing by the same nonzero number (any number except zero). In $x + 8 = 3$ , the number $8$ was added to $x$ , so you subtract $8$ . Subtract it from both sides, because subtracting the same number from both sides gives a new equation with the same solutions.

Then simplify each side. To simplify is to replace an expression with a simpler equivalent expression, one that has the same value for every permitted value of the variable. Here $x + 8 - 8$ is equivalent to $x$ , and $3 - 8 = -5$ .

x + 8 - 8 = 3 - 8 \quad\Longrightarrow\quad x = -5

Now check. Substitute $-5$ for $x$ in the original equation, which means replacing $x$ with $-5$ . You get $-5 + 8 = 3$ , a true statement, so $-5$ is a solution. Always check in the original equation. If you slip while solving, a later line may no longer have the same solutions as the equation you started with.

Only one operation had to be undone, so $x + 8 = 3$ is a one-step equation. The table shows a one-step equation for each of the four operations, with the inverse operation that solves it.

One-step equations: one inverse operation, done to both sides

Equation	Done to the variable	Inverse operation on both sides	Solution	Check in the original equation
$x + 8 = 3$	added $8$	subtract $8$	$x = 3 - 8 = -5$	$-5 + 8 = 3$ is true
$x - 4 = 11$	subtracted $4$	add $4$	$x = 11 + 4 = 15$	$15 - 4 = 11$ is true
$6x = -42$	multiplied by $6$	divide by $6$ (valid: $6 \neq 0$ )	$x = -42 \div 6 = -7$	$6(-7) = -42$ is true
$\frac{x}{5} = -2$	divided by $5$	multiply by $5$ (valid: $5 \neq 0$ )	$x = -2 \cdot 5 = -10$	$\frac{-10}{5} = -2$ is true

Key concept

Undo with inverse operations on both sides

To solve an equation, undo what was done to the variable with inverse operations, and do each one to both whole sides. Adding or subtracting the same number on both sides, or multiplying or dividing both sides by the same nonzero number, gives a new equation with exactly the same solutions. Never divide by zero. When the variable stands alone, check your value by substituting it into the original equation.

Worked example

A one-step equation with a negative multiplier

Solve $-3x = 7$ , where $x$ is a real number.

Read what was done to x: it was multiplied by -3. The inverse operation is dividing by -3. That step is valid because -3 is not zero.
$-3x = 7$
Divide both sides by -3. Dividing both sides by the same nonzero number gives a new equation with the same solutions.
$\frac{-3x}{-3} = \frac{7}{-3}$
Simplify each side. On the left, -3 divided by -3 is 1, so the left side is equivalent to x. On the right, a positive number divided by a negative number is negative.
$x = -\frac{7}{3}$
Check in the original equation by replacing x with this value. A negative number times a negative number is positive, and 21 divided by 3 is 7, so the equation is true.
$-3 \cdot \left(-\frac{7}{3}\right) = \frac{21}{3} = 7$
Keep the answer exact. The decimal -2.33 is only a rounded approximation of the fraction, and it does not make the original equation true.
$-3(-2.33) = 6.99 \neq 7$

Answer

The solution is $x = -\frac{7}{3}$ , the exact value that makes $-3x = 7$ true.

A two-step equation is one where two operations were done to the variable, so you need two inverse operations to solve it. Look at how the left side of $3x + 5 = 20$ was built: start with $x$ , multiply by $3$ , then add $5$ .

The most direct way back to $x$ is to undo those operations in reverse order, the way you take off your shoes before your socks. The last operation done is the first one you undo: subtract $5$ first, then divide by $3$ . The table plans three two-step equations this way. You will meet all three again below.

Plan a two-step equation: undo the last operation first

Equation	Done to the variable, in order	Undo, in reverse order
$3x + 5 = 20$	multiplied by $3$ , then added $5$	subtract $5$ , then divide by $3$
$\frac{x - 5}{2} = -4$	subtracted $5$ , then divided by $2$	multiply by $2$ , then add $5$
$\frac{x}{4} - 7 = -9$	divided by $4$ , then subtracted $7$	add $7$ , then multiply by $4$

Worked example

A first two-step equation

Solve $3x + 5 = 20$ , where $x$ is a real number.

Plan from how the left side was built: x was multiplied by 3, and then 5 was added. Undo in reverse order: subtract 5 first, then divide by 3.
Subtract 5 from both sides. Subtracting the same number from both sides keeps the solutions the same, and it leaves 3x alone on the left.
$3x + 5 - 5 = 20 - 5 \quad\Longrightarrow\quad 3x = 15$
Divide both sides by 3, which is valid because 3 is not zero. On the left, 3x divided by 3 is equivalent to x. On the right, 15 divided by 3 is 5.
$\frac{3x}{3} = \frac{15}{3} \quad\Longrightarrow\quad x = 5$
Check in the original equation by replacing x with 5.
$3(5) + 5 = 15 + 5 = 20$

Answer

The solution is $x = 5$ , because $3(5) + 5 = 20$ is true.

Worked example

A two-step equation with a fraction bar

Solve $\frac{x - 5}{2} = -4$ , where $x$ is a real number.

Plan: the fraction bar groups x - 5, so 5 was subtracted from x first, and then the result was divided by 2. The division came last, so undo it first: multiply by 2, then add 5.
Multiply both sides by 2, a nonzero number, so the solutions stay the same. On the left this undoes the division and leaves x - 5. On the right, a positive number times a negative number is negative.
$2 \cdot \frac{x - 5}{2} = 2 \cdot (-4) \quad\Longrightarrow\quad x - 5 = -8$
Add 5 to both sides to undo the subtraction.
$x - 5 + 5 = -8 + 5 \quad\Longrightarrow\quad x = -3$
Check in the original equation by replacing x with -3.
$\frac{-3 - 5}{2} = \frac{-8}{2} = -4$

Answer

The solution is $x = -3$ , because $\frac{-3 - 5}{2} = -4$ is true.

In the warning above, a check exposed a wrong answer. A check can also help you find the error, in your own work or in someone else's. Substitute the final value into the original equation. If the result is false, at least one step did not keep the solutions the same.

To locate that step, test the same value in each line of the work. Every correct step keeps the solutions the same, so a value that makes one line false must make the next line false too. When the value makes a line false but makes the very next line true, the error is in the step between them. Look at that step and ask: was the same valid operation done to both whole sides, with correct arithmetic?

Worked example

Find and correct an error

A student solved $\frac{x}{4} - 7 = -9$ in three lines. Line 1: $\frac{x}{4} - 7 = -9$ . Line 2: $\frac{x}{4} = -16$ . Line 3: $x = -64$ . Find the error and correct it. Here $x$ is a real number.

Check the student's value in the original equation, Line 1, by replacing x with -64. The left side comes to -23, not -9, so -64 is not a solution and the work contains an error.
$\frac{-64}{4} - 7 = -16 - 7 = -23 \neq -9$
Test -64 in Line 2. The line is true. Since -64 makes Line 1 false and Line 2 true, the step from Line 1 to Line 2 changed the solutions, so the error is in that step.
$\frac{-64}{4} = -16$
Name the error. On the left, the subtraction of 7 disappeared, which happens when 7 is added to that side. On the right, -9 became -16, which means 7 was subtracted. The two sides got different operations, so the solutions were not kept.
$-9 - 7 = -16 \quad\text{but}\quad -9 + 7 = -2$
Correct the step by adding 7 to both sides, which keeps the solutions the same.
$\frac{x}{4} - 7 + 7 = -9 + 7 \quad\Longrightarrow\quad \frac{x}{4} = -2$
Multiply both sides by 4, a nonzero number, to undo the division. A positive number times a negative number is negative.
$4 \cdot \frac{x}{4} = 4 \cdot (-2) \quad\Longrightarrow\quad x = -8$
Check the corrected value in the original equation by replacing x with -8.
$\frac{-8}{4} - 7 = -2 - 7 = -9$

Answer

The student subtracted $7$ on the right side instead of adding $7$ to both sides. The correct solution is $x = -8$ , because $\frac{-8}{4} - 7 = -9$ is true.

Takeaway. To solve an equation, undo what was done to the variable. Use inverse operations, starting with the last operation that was done, and do each step to both whole sides, never dividing by zero. Then substitute your value into the original equation. A true statement confirms the solution. A false one tells you to test each line and find the step that changed the solutions. Next, you will model and solve equations using algebra tiles.