Solving One-Step Equations

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An equation is a statement of equality. The quantity on the left-hand side is equal to the quantity on the right-hand side. Solving an equation means finding the value of the variable that makes the statement true. This is done by isolating the variable on one side of the equation, through the use of inverse operations and the Properties of Equality.
Concept

Inverse Operations

Inverse operations are two operations that, all else being equal, undo one another. For instance, adding 66 and subtracting 66 are inverse operations because they cancel each other out. x+6x+6  6x\begin{aligned} &x+6\\ &x+6{\color{#0000FF}{\ -\ 6}}\\ &x \end{aligned} In an expression like x+6,x+6, the addition of 66 to xx is eliminated by performing the inverse operation: a subtraction of 6.6. Using inverse operations on an equation, however, is a little different: in order to adhere to the Properties of Equality, any operation performed on one side of an equation must also be performed on the other side to maintain equality. x÷2=1x÷2 × 2=1 × 2x=2\begin{aligned} x\div{2}&=1\\ x\div{2}{\color{#0000FF}{\ \times{\ 2}}}&=1{\color{#0000FF}{\ \times{\ 2}}}\\ x&=2 \end{aligned}

In this case, the division by 22 on one side of the equation could only be eliminated by a multiplication by 22 on both sides of the equation. The result of applying the Properties of Equality on an equation is an equivalent equation.
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Addition and Subtraction Properties of Equality

The Addition Property of Equality states that adding the same number to each side of an equation yields an equivalent equation.

If a=b,a = b, then a+c=b+c.a + c = b + c.

The Subtraction Property of Equality is similar to the Addition one, with the exception being the sign used — subtracting the same number from each side of an equation yields an equivalent equation.

If a=b,a = b, then ac=bc.a - c = b - c.

Either of these properties can be used when eliminating a term in an equation. Deciding which to use depends on if the term is being added to or subtracted from the variable.

Misc Addition and Subtraction Property of Equality 1.svg
Exercise

Solve the following equations. x4=9andx+5=3 x - 4 = 9 \quad \text{and} \quad x + 5 = 3

Solution
Example

x4=9x - 4 = 9

To solve this equation, we have to isolate the variable x.x. For this, we can eliminate the term being subtracted from xx by performing the inverse operation. Since the inverse operation of subtracting 44 is adding 4,4, we'll add 44 to each side of the equation. In line with the Addition Property of Equality, this produces an equivalent equation.

x4=9x - 4 = 9
x4+4=9+4x - 4 + 4 = 9 + 4
x=13x = 13

Thus, the solution to the equation is x=13.x = 13.

Example

x+5=3x + 5 = 3

Here, the +5+5 has to be eliminated for us to isolate x.x. The inverse operation of adding 55 is subtracting 5.5. Thus, we'll subtract 55 from each side of the equation. By the Subtraction Property of Equality, this yields an equivalent equation.

x+5=3x + 5 = 3
x+55=35x + 5 - 5 = 3 - 5
x=-2x = \text{-} 2

x=-2x = \text{-} 2 is the solution of the equation.

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Concept

Multiplication and Division Properties of Equality

The Multiplication Property of Equality states that multiplying each side of an equation by the same number yields an equivalent equation.

If a=b,a = b, then ac=bc.a \cdot c = b \cdot c.

Similarly, by the Division Property of Equality, an equivalent equation is produced when each side of an equation is divided by the same non-zero number.

If a=ba = b and c0,c \neq 0, then ac=bc.\dfrac{a}{c} = \dfrac{b}{c}.

These two properties can be used when solving equations to eliminate denominators and factors, since multiplication is the inverse operation of division and vice versa.

Misc Multiplication and Division Property of Equality 1.svg
Exercise

Solve the following equations. a3=7and11b=22. \dfrac{a}{3} = 7 \quad \text{and} \quad 11b = 22.

Solution
Example

a3=7\dfrac{a}{3} = 7

To isolate the variable a,a, we have to eliminate the denominator 3.3. For this we can use the inverse operation of division, which is multiplication. According to the Multiplication Property of Equality, multiplying each side of the equation by 33 will produce an equivalent equation. Notice that it will also eliminate the denominator.

a3=7\dfrac{a}{3} = 7
a33=73\dfrac{a}{3} \cdot 3 = 7 \cdot 3
a=21a = 21

We have found that a=21a = 21 solves the equation.

Example

11b=2211b = 22

Since division is the inverse operation of multiplication, we'll use the Division Property of Equality to eliminate the factor 11.11. This is done by dividing each side of the equation by 11.11.

11b=2211b = 22
11b11=2211\dfrac{11b}{11} = \dfrac{22}{11}
b=2b = 2

The solution of the equation is b=2.b = 2.

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Exercises

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