Equivalent Expressions and Error Analysis | Grade 7 Math | LevelKite

Two expressions can look different and still give the same result every time. In this lesson you will decide when that is true, and you will find and fix mistakes in worked-out algebra.

First, the words. An expression is a mathematical phrase built from numbers, variables and operations, such as $2(x + 3)$ . A variable is a letter that stands for a number. In this lesson, $x$ stands for any number, including negative numbers, zero, fractions and decimals. A linear expression is one that can be written as a number times the variable plus a number, such as $2x + 6$ .

Two expressions are equivalent only if they have the same value for every value of the variable. Being equal for some values is not enough. The table below tests three expressions with the same five values of $x$ .

Three expressions evaluated at the same five values of x

Value of x	2(x + 3)	2x + 6	2x + 3
$-2$	$2(-2 + 3) = 2(1) = 2$	$2(-2) + 6 = -4 + 6 = 2$	$2(-2) + 3 = -4 + 3 = -1$
$0$	$2(0 + 3) = 2(3) = 6$	$2(0) + 6 = 0 + 6 = 6$	$2(0) + 3 = 0 + 3 = 3$
$\frac{1}{2}$	$2(\frac{1}{2} + 3) = 2(\frac{7}{2}) = 7$	$2(\frac{1}{2}) + 6 = 1 + 6 = 7$	$2(\frac{1}{2}) + 3 = 1 + 3 = 4$
$1$	$2(1 + 3) = 2(4) = 8$	$2(1) + 6 = 2 + 6 = 8$	$2(1) + 3 = 2 + 3 = 5$
$5$	$2(5 + 3) = 2(8) = 16$	$2(5) + 6 = 10 + 6 = 16$	$2(5) + 3 = 10 + 3 = 13$

Read across each row. The third expression, $2x + 3$ , never matches $2(x + 3)$ in this table. You need only one row to rule it out: at $x = 0$ , $2(x + 3)$ is $6$ but $2x + 3$ is $3$ . Equivalent expressions must agree for every value of $x$ , so one mismatch proves that $2(x + 3)$ and $2x + 3$ are not equivalent.

The first two expressions agree in all five rows. That is good evidence, but a table can hold only a few values of $x$ , and equivalence is a claim about every value. To be sure, you need a reason that works for all numbers at once. The distributive property gives one: multiplying a sum by a number gives the same result as multiplying each part of the sum by that number and then adding the products. It is true whatever number $x$ is, so it rewrites $2(x + 3)$ without changing its value:

2(x + 3) = 2 \cdot x + 2 \cdot 3 = 2x + 6 \quad \text{for every value of } x

Key concept

Equivalent means equal for every value

Two expressions are equivalent only if they have the same value for every value of the variable.

To show that two expressions are equivalent, rewrite them using properties that are true for all numbers, such as the distributive property. If they become the same expression, they are equivalent.
To show that two expressions are not equivalent, find one value of the variable that gives different results. One mismatch is enough.

A value that gives matching results tells you only about that value. On its own, it does not show that the expressions are equivalent.

Here is a dependable way to compare two linear expressions. First simplify each one: rewrite it as an equivalent expression with no parentheses and no like terms left to combine.

Expand. Use the distributive property to multiply out any parentheses.
Combine like terms. Terms are the parts of an expression joined by addition or subtraction, and each term keeps the sign in front of it. Like terms have the same variable part, such as $4x$ and $2x$ , or are both plain numbers (called constants), such as $-8$ and $5$ . Combining them is the distributive property used in reverse, for example $4x + 2x = (4 + 2)x = 6x$ , so it keeps the value the same for every value of $x$ .

Then compare the two simplified expressions term by term. They are equivalent exactly when the $x$ terms are the same and the plain numbers are the same. A missing term counts as $0x$ or as $0$ . If either part is different, the expressions are not equivalent, and you can find a value of $x$ that shows the mismatch.

Worked example

Which expressions are equivalent?

Which of these expressions are equivalent to $4(x - 2) + 2x$ ? A is $6x - 8$ . B is $2(3x - 4)$ . C is $6x - 2$ . Here $x$ stands for any number.

Simplify the given expression. Start by expanding the parentheses with the distributive property: multiply 4 by each term inside, x and -2. This property is true for every value of x, so the value of the expression does not change.
$4(x - 2) + 2x = 4 \cdot x + 4 \cdot (-2) + 2x = 4x - 8 + 2x$
Each term keeps its sign, and terms can be added in any order, so bring the like terms 4x and 2x together: 4 + 2 = 6 copies of x. The plain number -8 has no like term, so it stays as it is.
$4x - 8 + 2x = (4 + 2)x - 8 = 6x - 8$
Compare with A. The simplified form is exactly A. Every rewriting step is true for all numbers, so A has the same value as the given expression for every value of x. A is equivalent.
$4(x - 2) + 2x = 6x - 8$
Now simplify B by expanding it: multiply 2 by each term inside, 3x and -4. The result is the same simplified form, so B is also equivalent to the given expression.
$2(3x - 4) = 2 \cdot 3x + 2 \cdot (-4) = 6x - 8$
Compare with C. Its x term matches, but its plain number is -2 instead of -8, so look for a mismatch. Test x = 0 in the given expression and in C. The results, -8 and -2, are different. One mismatch is enough to show that C is not equivalent.
$\text{given: } 4(0 - 2) + 2(0) = 4(-2) + 0 = -8 \qquad \text{C: } 6(0) - 2 = 0 - 2 = -2$

Answer

A and B are both equivalent to $4(x - 2) + 2x$ , because all three simplify to $6x - 8$ and so are equal for every value of $x$ . C is not equivalent: at $x = 0$ it gives $-2$ , but the given expression gives $-8$ .

Now use the same idea to find mistakes. Error analysis means reading worked-out steps, finding where they go wrong, explaining why, and fixing them.

When an expression is simplified correctly, every line is equivalent to the line before it. So to find a mistake, look for a line that is not equivalent to the line before it:

Compare each line with the line just above it, one step at a time.
Ask which property the step was meant to use, and redo that step yourself.
If your result is not equivalent to theirs, the step is wrong. One value of $x$ that gives different results in the two versions proves it.
Fix the step, then finish the work from there.

A later line can follow correctly from a wrong line. That later step is not a new mistake, but its result is still not equivalent to the original expression.

Common mistakes when expanding and simplifying. Read each arrow as: was rewritten as.

Mistaken rewrite	What went wrong	Correct rewrite	A test value that shows the mismatch
$2(x + 3) \rightarrow 2x + 3$	The $2$ multiplied $x$ but not $3$ . It must multiply every term inside the parentheses.	$2 \cdot x + 2 \cdot 3 = 2x + 6$	$x = 0$ : $2(0 + 3) = 6$ , but $2(0) + 3 = 3$
$-3(x - 2) \rightarrow -3x - 6$	A negative times a negative is positive, so $(-3)(-2) = +6$ . The sign was lost.	$(-3)(x) + (-3)(-2) = -3x + 6$	$x = 0$ : $-3(0 - 2) = 6$ , but $-3(0) - 6 = -6$
$7x - (2x + 5) \rightarrow 7x - 2x + 5$	The subtraction applies to every term in the parentheses. Only $2x$ was subtracted; the $5$ was added.	$7x - 2x - 5 = 5x - 5$	$x = 0$ : $7(0) - (2(0) + 5) = -5$ , but $7(0) - 2(0) + 5 = 5$
$3x + 4 \rightarrow 7x$	$3x$ and $4$ are not like terms, so they cannot be combined into one term.	None needed: $3x + 4$ is already simplified.	$x = 0$ : $3(0) + 4 = 4$ , but $7(0) = 0$
$6x + 12 \rightarrow x + 2$	Dividing each term by $6$ gives an expression with one-sixth of the value. Keep the $6$ outside.	$6 \cdot x + 6 \cdot 2 = 6(x + 2)$	$x = 1$ : $6(1) + 12 = 18$ , but $1 + 2 = 3$

Worked example

Find and fix a sign mistake

Lena simplified $5x - 2(x - 4)$ , where $x$ stands for any number. Her Line 1 is $5x - 2x - 8$ . Her Line 2 is $3x - 8$ . Find her mistake, explain it, and write a correct simplified expression.

Compare Line 1 with the original expression. Subtracting 2(x - 4) is the same as adding -2(x - 4), so redo that expansion: multiply -2 by each term inside, x and -4. A negative times a negative is positive, so (-2)(-4) = +8. This fits the meaning: 2(x - 4) is 8 less than 2x, so taking it away leaves 8 more than taking away 2x.
$-2(x - 4) = (-2)(x) + (-2)(-4) = -2x + 8$
Lena wrote -8 where the expansion gives +8. She multiplied 2 by 4 and copied the minus sign from inside the parentheses, instead of multiplying -2 by -4. Test x = 0 in the original expression and in her Line 1 to confirm.
$\text{original: } 5(0) - 2(0 - 4) = 0 - 2(-4) = 0 + 8 = 8 \qquad \text{Line 1: } 5(0) - 2(0) - 8 = -8$
The results, 8 and -8, are different, so Line 1 is not equivalent to the original expression. Write the corrected Line 1, using the expansion from the first step.
$5x - 2(x - 4) = 5x - 2x + 8$
Combine the like terms 5x and -2x. That makes 5 - 2 = 3 copies of x. The plain number +8 has no like term, so it stays. Lena's Line 2 combined like terms correctly; it is wrong only because it started from a wrong Line 1.
$5x - 2x + 8 = (5 - 2)x + 8 = 3x + 8$
Check the arithmetic with another value, x = 1. The original expression and the corrected answer both give 11. They must match, because each corrected step uses a property that is true for every value of x.
$\text{original: } 5(1) - 2(1 - 4) = 5 - 2(-3) = 5 + 6 = 11 \qquad \text{corrected: } 3(1) + 8 = 11$

Answer

Lena's mistake is in Line 1: $-2$ times $-4$ is $+8$ , not $-8$ . The correct simplified expression is $3x + 8$ , which is equivalent to $5x - 2(x - 4)$ for every value of $x$ .

Worked example

Find two mistakes in one solution

Ravi simplified $\frac{1}{2}(8x - 6) + 5 - x$ , where $x$ stands for any number. His Line 1 is $4x - 6 + 5 - x$ . His Line 2 is $3x - 1$ . His Line 3 is $2x$ . Find every mistake and write a correct simplified expression.

Compare Line 1 with the original expression. Expanding means multiplying one-half by both terms inside the parentheses, 8x and -6. Half of -6 is -3, but Ravi kept -6, so the one-half never reached the second term. This is the first mistake. At x = 0, the correct expansion gives -3 and Ravi's gives -6.
$\frac{1}{2}(8x - 6) = \frac{1}{2} \cdot 8x + \frac{1}{2} \cdot (-6) = 4x - 3$
Compare Line 2 with Line 1. Ravi combined the x terms, 4x - x = 3x (remember that x means 1x), and the plain numbers, -6 + 5 = -1. Those are like terms and the arithmetic is right, so this step is valid. It carries the first mistake forward but adds no new one.
$4x - 6 + 5 - x = (4 - 1)x + (-6 + 5) = 3x - 1$
Compare Line 3 with Line 2. Ravi turned 3x - 1 into 2x, but 3x and -1 are not like terms: 3x is three copies of x, and -1 is a plain number. At x = 0, Line 2 gives -1 and Line 3 gives 0, so this is the second mistake. At x = 1 both lines give 2, which shows how a single test value can hide a mistake.
$\text{Line 2: } 3(0) - 1 = -1 \qquad \text{Line 3: } 2(0) = 0$
Start again from the original expression and expand correctly, using the result from the first step.
$\frac{1}{2}(8x - 6) + 5 - x = 4x - 3 + 5 - x$
Combine like terms. The x terms make 4 - 1 = 3 copies of x, and the plain numbers make -3 + 5 = 2. The terms 3x and 2 are not like terms, so stop here.
$4x - 3 + 5 - x = (4 - 1)x + (-3 + 5) = 3x + 2$
Check the arithmetic at x = 2. The original expression gives 8, as shown. The corrected answer gives 3(2) + 2 = 8, which matches, and Ravi's answer gives 2(2) = 4, which does not. The match checks your arithmetic. The properties used in the last two steps are what make the expressions equal for every value of x.
$\frac{1}{2}(8 \cdot 2 - 6) + 5 - 2 = \frac{1}{2}(10) + 5 - 2 = 5 + 5 - 2 = 8$

Answer

Ravi made two mistakes: in Line 1 he did not multiply $-6$ by $\frac{1}{2}$ , and in Line 3 he combined $3x$ and $-1$ , which are not like terms. The correct simplified expression is $3x + 2$ , which is equivalent to $\frac{1}{2}(8x - 6) + 5 - x$ for every value of $x$ .

The big idea: two expressions are equivalent only if they are equal for every value of the variable. To show that they are equivalent, rewrite them with properties that are true for all numbers until they are the same expression. To show that they are not equivalent, find one value of the variable that gives different results.

Error analysis uses the same idea. In correct work, each line is equivalent to the line before it, so a mistake is where that chain breaks. Find that line, say what went wrong, and repair it.