Linear Equation

Sommaire

In short: Linear equation names an equation where the variable appears only to the first power, so it has the shape of a straight line and, in one variable, one solution, none, or infinitely many.

8th GradeAlgebra8.EE.C.7

What is Linear Equation?

Whenever the unknown shows up only as itself, never squared or under a root, the statement is a linear equation. It can hold the variable on both sides, as in $3x + 19 = 7x – 9$, and it can be solved by balancing: whatever you do to one side, do to the other.

After simplifying, a one-variable case lands in one of three outcomes. It reaches $x = \text{number}$, a single solution; it reaches a false statement like $5 = 8$, no solution; or it reaches a true statement like $7 = 7$, where every number works.

How to spot it

A linear equation contains x only to the first power, with no x squared and no root of x.
Variable terms appear on both sides and must be gathered together before dividing.
A word problem says two plans, prices or quantities become equal after some unknown amount of time.
Distributing a number across parentheses is needed before the sides can be compared.

The method

To solve a linear equation, clear parentheses by distributing, then combine like terms on each side separately.
Move all variable terms to one side and all plain numbers to the other by adding or subtracting.
Divide both sides by the coefficient of the variable.
Substitute your answer into the original equation to check; a false check means a slip somewhere.

Both plans cost $40 after 7 months, where the two lines meet.

Linear Equation: solved example

Problem

Plan A charges $\$19$ to join plus $\$3$ per month. Plan B charges $\$7$ per month but gives a one-time $\$9$ rebate. After how many months do the two plans cost the same?

Solution

Set the costs equal: $3x + 19 = 7x – 9$. Subtract $3x$ from both sides: $19 = 4x – 9$. Add 9: $28 = 4x$.

Divide by 4: $x = 7$. Check: $3(7) + 19 = 40$ and $7(7) – 9 = 40$. The plans match after 7 months, at $\$40$.

Try it

Question 1

Solve $2(x + 5) = 3x – 4$.

Show the answer

$2x + 10 = 3x – 4$, so $14 = x$. Check: $2(19) = 38$ and $3(14) – 4 = 38$.

Question 2

How many solutions does $4x + 6 = 4x + 11$ have?

Show the answer

Subtracting $4x$ leaves $6 = 11$, which is false, so there is no solution.

Linear Equation vs. Linear Function

An equation is a statement with an equals sign that you solve for the unknown. A function is a rule that sends every input to one output and is graphed or evaluated. Setting a function’s output equal to a number turns it into a question that you solve, which is how the two ideas meet.

Linear Function

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Whole dictionary: A–Z · 8th Grade · Algebra

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