What Makes a Value a Solution of an Equation? | Grade 6 Math | LevelKite

You have used the equals sign for years. Now look closely at what it claims. An equation is a statement that says two expressions have the same value: the left side and the right side are the same number.

When an equation has only numbers, you can check the claim right away. The equation $4 + 5 = 9$ is true, because both sides are $9$ . The equation $4 + 5 = 10$ is false, because the left side is $9$ and the right side is $10$ .

Now put a variable in the equation. A variable is a letter that stands for a number. Whether the claim is true can now depend on which number the letter stands for. So the equation asks a question: which numbers make this claim true?

Here is an example. In it, the letter $x$ stands for a number.

x + 5 = 12

The left side of this equation is the expression $x + 5$ . The right side is $12$ .

To test a number, substitute it: replace the variable with that number. Then find the value of each side and see whether the two sides match. The table tests four numbers. In the table, the symbol $\neq$ means is not equal to.

Testing values of x in the equation x + 5 = 12

Value of x	Left side: x + 5	Right side	Is the equation true?
$5$	$5 + 5 = 10$	$12$	No, because $10 \neq 12$
$6$	$6 + 5 = 11$	$12$	No, because $11 \neq 12$
$7$	$7 + 5 = 12$	$12$	Yes, because $12 = 12$
$8$	$8 + 5 = 13$	$12$	No, because $13 \neq 12$

Only $7$ makes the two sides match. Each of the other numbers in the table gives a left side that is too small or too large.

No number outside the table works either. A number less than $7$ gives a sum less than $12$ , and a number greater than $7$ gives a sum greater than $12$ . So $7$ is the one number that makes $x + 5 = 12$ true.

Sometimes the question comes with a list of allowed values. If you could choose only from $5$ , $6$ and $8$ , the answer would be that none of them makes the equation true.

Key concept

A solution makes the equation true

A solution of an equation is a value of the variable that makes the equation true. When you substitute it everywhere the variable appears, the left side and the right side have the same value. You can also say that the value satisfies the equation.

To solve an equation is to answer its question: out of the values the variable is allowed to take, which ones make the equation true? There may be one such value, more than one, or none.

You ran a test four times in the table. Here it is as three steps, called the substitution check. Use it to decide whether a value is a solution.

Substitute. Replace the variable with the value everywhere it appears, on both sides.
Find the value of each side. Follow the order of operations. Use only valid operations, which are operations that give a number as a result. In particular, you can divide only by a nonzero quantity, which means a number that is not zero.
Compare. If both sides are the same number, the value is a solution. If they are different numbers, it is not a solution. If a side has no value, because finding it would mean dividing by zero, the value you tested is not a solution either.

The check answers one question: does this value make the equation true?

Worked example

Check one value

Let $n$ stand for a number. Is $n = 9$ a solution of $4n = 36$ ?

Substitute. Replace n with 9 on the left side. Here 4n means 4 times n, so it becomes 4 times 9, not 49. The raised dot in this step's expression means times. The right side has no n, so it stays 36.
$\text{Left side: } 4 \cdot 9 \qquad \text{Right side: } 36$
Multiply to find the value of the left side, so that you can compare it with the right side.
$4 \cdot 9 = 36$
Compare. The left side is 36 and the right side is 36. They are the same number, so the equation is true when n is 9.
$36 = 36$

Answer

Yes, $n = 9$ is a solution of $4n = 36$ , because substituting $9$ for $n$ makes both sides equal $36$ .

Worked example

Test two values in the same equation

Let $x$ stand for a number. Is $x = 5$ a solution of $3x + 4 = 22$ ? Is $x = 6$ a solution?

Start with 5. Replace x with 5 on the left side, so 3x becomes 3 times 5. The right side has no x, so it stays 22.
$\text{Left side: } 3 \cdot 5 + 4 \qquad \text{Right side: } 22$
Find the value of the left side. The order of operations says to multiply first and then add.
$3 \cdot 5 + 4 = 15 + 4 = 19$
Compare. The left side is 19 and the right side is 22. They are different numbers, so the equation is false when x is 5.
$19 \neq 22$
Now test 6 the same way. Replace x with 6, multiply, and then add.
$3 \cdot 6 + 4 = 18 + 4 = 22$
Compare. Both sides are 22, so the equation is true when x is 6.
$22 = 22$

Answer

The value $x = 5$ is not a solution of $3x + 4 = 22$ , because it makes the left side $19$ while the right side is $22$ . The value $x = 6$ is a solution, because it makes both sides equal $22$ .

The check uses the same value everywhere the variable appears, on both sides. Treating both sides alike also keeps a true equation true. Picture a level balance scale: add the same weight to both pans and it stays level.

Take the true equation $7 + 5 = 12$ . Add $3$ to both sides and you get $7 + 5 + 3 = 12 + 3$ . Both sides are $15$ , so the equation is still true. Add $3$ to the left side only and you get $7 + 5 + 3 = 12$ . That says $15$ equals $12$ , which is false.

So here is the rule, with its conditions. Two equal sides stay equal when you apply the same valid operation to both sides. Division needs extra care: you can divide only by a nonzero quantity, because dividing by zero does not give a number. That condition matters in a substitution check too, as the next example shows.

Worked example

When a value would mean dividing by zero

Let $n$ stand for a number. Is $n = 4$ a solution of $\frac{20}{n} + 1 = 6$ ? Is $n = 0$ a solution?

Start with 4. Replace n with 4 on the left side. The fraction bar means divide, so the left side is 20 divided by 4, plus 1. The right side has no n, so it stays 6.
$\text{Left side: } \frac{20}{4} + 1 \qquad \text{Right side: } 6$
Find the value of the left side. Divide first. You are dividing by 4, which is not zero, so the division is valid. Then add.
$\frac{20}{4} + 1 = 5 + 1 = 6$
Compare. Both sides are 6, so the equation is true when n is 4.
$6 = 6$
Now test 0. Replace n with 0 on the left side, just as you did with 4.
$\frac{20}{0} + 1$
Stop here. You would be dividing by 0, and you can divide only by a nonzero quantity. No number times 0 equals 20, so 20 divided by 0 has no value. The left side has no value to compare with 6, so the equation is not true when n is 0.

Answer

The value $n = 4$ is a solution of $\frac{20}{n} + 1 = 6$ , because it makes both sides equal $6$ . The value $n = 0$ is not a solution, because it would mean dividing by zero, so the left side has no value.

Here is the idea to keep. An equation with a variable asks a question: which values make it true? A solution is a value that does. To solve the equation is to find every allowed value that does.

You test a value by substituting it everywhere the variable appears and comparing the two sides. The same number on both sides means the value is a solution. Different numbers, or a side with no value, means it is not.

Keep the conditions with the test. Equal sides stay equal when you apply the same valid operation to both sides, and you can divide only by a nonzero quantity. One value that works speaks only for itself. It does not make the two sides equivalent expressions.

Next, you will use the same substitution check on an inequality, a statement that compares two sides with a phrase such as is less than or is greater than. You will also find which values in a set make a statement true.