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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

Your turn!

TRY: ONE-STEP LINEAR EQUATION
If x, plus, 2, equals, 9, what is the value of x ?
x, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

TRY: TWO-STEP LINEAR EQUATION
If 3, x, minus, 5, equals, 35, what is the value of x ?
Choose 1 answer:
Choose 1 answer:

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?
Choose 1 answer:
Choose 1 answer:

TRY: LINEAR EQUATION WITH FRACTION COEFFICIENT
If start fraction, 1, divided by, 6, end fraction, x, plus, 3, equals, 7, what is the value of x ?
x, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Things to remember

To solve a linear equation, we find the value of the variable that makes the equation true by:
  1. Distributing any coefficients.
  2. Combining any like terms.
  3. Isolating the variable.

Want to join the conversation?

  • blobby green style avatar for user SummerKiley
    I Just failed my quiz for school and this really is not helping on performing operations on linear equations and I can't ask my siblings, and I can't find any thing on here to help me! Somebody help me!
    (3 votes)
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  • blobby green style avatar for user saksham.kapoor2011
    Wow i had a test on this today and i gto a really good score
    (2 votes)
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  • blobby green style avatar for user tjsimmons02
    I'm more confused after watching this one.
    (2 votes)
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  • blobby green style avatar for user saksham.kapoor2011
    And what app do you use i really need it
    (1 vote)
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  • blobby green style avatar for user frich41
    what should I do next? I am scheduled to take the praxis 8/19/22.
    (1 vote)
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  • blobby green style avatar for user KEVINW
    In the tips section, you clearly state to you the inverse operation to isolate variables. Yet, in your last example, you use the same operation, instead of the inverse. How does this help? It confuses.
    (1 vote)
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