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Concept

Solution - Equation

A solution of an equation is a value that makes the equation true. It satisfies the equation when it is substituted for the variable of the equation. Consider the following equation.

x + 1 = 4

The equation has the solution

x = 3

because

3

is the only value of

x

that makes the equation true. This means that the the right and left-hand sides of the equation are equal to each other when this solution is substituted for the variable.

x + 1 = 4

Substitute

$x = 3$

3 + 1 = ? 4

AddTerms

Add terms

4 = 4 ✓

For the other values that are not solutions, the right-hand side of the solution is not equal to the left-hand side. This is shown by a not equal sign

\neq = .

Consider the result of substituting

x = 1

into the equation.

x + 1 = 4

Substitute

$x = 1$

1 + 1 = ? 4

AddTerms

Add terms

2 \neq = 4 \times

Some equations are impossible to solve. Consider the following example.

x = x + 1

Notice that it is impossible to get a number itself after adding

1

to that number. This means that this equation has no solution. Conversely, it is possible for an equation to have more than one solution. An equation can also have infinitely many solutions, which means that all values satisfy the equation.

x + x = 2 x

This equation will be true for all values of

x

since the expressions on the left-hand side and right-hand side are equivalent expressions.

Extra

Replacement Set

Sometimes the solution set of an equation can be obtained from a replacement set, which is a list of possible solutions. The values from the replacement set are substituted into the equation and then checked to confirm whether they satisfy the equation or not. Consider the following equation and its replacement set.

Equation : Replacement Set : x - 5 = 3 {5, 6, 7, 8}

In the following table, each value is substituted into the equation and the equation is evaluated.

Value of $x$	Substitute	Are both sides equal?
$5$	$5 - 5 = ? 3$	$0 \neq = 3 \times$
$6$	$6 - 5 = ? 3$	$1 \neq = 3 \times$
$7$	$7 - 5 = ? 3$	$2 \neq = 3 \times$
$8$	$8 - 5 = ? 3$	$3 = 3 ✓$

As shown in the table, the only solution of the equation is $3 .$

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