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# Linear equations | Lesson

## What are linear equations?

A linear equation is an equation with
variable(s) to the first power
and one or more constants. For example, in the linear equation start color #11accd, 2, end color #11accd, x, plus, start color #ca337c, 3, end color #ca337c, equals, start color #ca337c, 4, end color #ca337c:
• x is the variable, which represents a number whose value we don't know yet.
• start color #11accd, 2, end color #11accd is the coefficient, or the constant multiple of the variable x. Together, start color #11accd, 2, end color #11accd, x is a single
term
.
• start color #ca337c, 3, end color #ca337c and start color #ca337c, 4, end color #ca337c are constants. They are both terms as well.
Solving a linear equation means finding the value(s) for the variable(s) that make the equation true.

### What skills are tested?

• Solving linear equations with one variable

## How do we solve linear equations?

Our goal when solving linear equations with one variable is to find the value of the variable that makes the equation true. To do this, we need to
isolate the variable
by identifying
inverse operations
to perform on both sides of the equation until the variable is left by itself.
• Addition and subtraction are inverse operations.
• Multiplication and division are inverse operations.
For
two-step linear equations
, it's easiest if we first combine the constant terms on one side of the equation and the x-terms on the other side of the equation. Then, isolate x.

## What features make linear equations more difficult?

Below are features that make solving linear equations more challenging and tips for handling them.

### Negative numbers

When working with negative numbers, remember that:
• start text, n, e, g, a, t, i, v, e, end text, times, start text, n, e, g, a, t, i, v, e, end text, equals, start text, p, o, s, i, t, i, v, e, end text
• start text, p, o, s, i, t, i, v, e, end text, times, start text, n, e, g, a, t, i, v, e, end text, equals, start text, n, e, g, a, t, i, v, e, end text

### Fractions

When solving a linear equation with fraction coefficients or constants:
• If the equation has only a fraction coefficient, consider leaving the fraction until the last step in isolating x.
• If the equation has both fraction coefficients and fraction constants, consider getting rid of the fractions in the first step.

### More than one variable term

When solving a linear equation with multiple terms containing the variable, we need to combine
like terms
. Just as constants can be added and subtracted, like terms can be combined by adding and subtracting the coefficients and keeping the variable the same:
a, x, plus minus, b, x, equals, left parenthesis, a, plus minus, b, right parenthesis, x

### Coefficients to distribute

When distributing coefficients, observe the distributive property:
a, left parenthesis, b, x, plus minus, c, right parenthesis, equals, a, b, x, plus minus, a, c

TRY: ONE-STEP LINEAR EQUATION
If x, plus, 2, equals, 9, what is the value of x ?
x, equals

TRY: TWO-STEP LINEAR EQUATION
If 3, x, minus, 5, equals, 35, what is the value of x ?

TRY: DISTRIBUTING AND COMBINING LIKE TERMS
2, left parenthesis, 3, x, minus, 1, right parenthesis, equals, 7, x, minus, 3
What is the solution to the equation above?