Linear Equations – Definition and How to Solve

Solving equations is probably the most important and useful skill to be learned in algebra. In this article, we will look at the definition of linear equations. We will also learn how to solve these equations by using a process and looking at solved exercises to improve our understanding.

ALGEBRA
linear equations - definition and how to solve

Relevant for…

Learning how to solve linear equations.

See examples

ALGEBRA
linear equations - definition and how to solve

Relevant for…

Learning how to solve linear equations.

See examples

Definition of linear equations

Linear equations are defined as the equations in which the maximum power of the variables is 1. These equations are called linear because they produce a straight line when graphed.

The following are examples of linear equations and nonlinear equations:

EXAMPLES

Linear equation: 

  • 4x+2=10
  • \frac{1}{3}x=9
  • 3x+2=-2x+1
  • 4(x-2)+2=-2x+1

Nonlinear equation: 

  • {{x}^{2}}+x-3=2
  • 2{{x}^{2}}-3{{x}^{2}}=6
  • \frac{1}{2}{{x}^{2}}=2x+1

How to solve linear equations?

To solve linear equations, we have to remember that we can perform any operation on one side of the equation, as long as we perform the same operation on the opposite side of the equation.

The following steps are a good method we can use to solve equations:

1. Simplify each side of the equation by removing parentheses and combining like terms.

2. Use addition and subtraction to solve for terms with variables on one side of the equation.

3. Use multiplication and division to solve the equation.


Linear equations – Examples with answers

EXAMPLE 1

  • Solve the equation 3x+5=14 for x.

Solution:

1. Simplify: it is already simplified.

2. Solve for the variables: we move the 5 to the right:

3x+5-5=14-5

3x=9

3. Solve the equation: we divide by 3:

\frac{3}{3}x=\frac{9}{3}

x=3

EXAMPLE 2

  • Solve the equation 3(x-5)-x=-7 for x.

Solution:

1. Simplify:

3(x-5)-x=-7

3x-15-x=-7

2x-15=-7

2. Solve for the variables: we move the -15 to the right:

2x-15+15=-7+15

2x=8

3. Solve the equation: we divide by 2:

\frac{2}{2}x=\frac{8}{2}

x=4

EXAMPLE 3

  • Solve 3(x-1)+2x=x+5 for x.

Solution:

1. Simplify:

3(x-1)+2x=x+5

3x-3+2x=x+5

5x-3=x+5

2. Solve for the variables: we move the -3 to the right and the x to the left:

5x-3+3=x+3+5

5x=x+8

5x-x=x+8-x

4x=8

3. Solve the equation: we divide by 4:

\frac{4}{4}x=\frac{8}{4}

x=2

EXAMPLE 4

  • Solve the equation 4(x-2)+5=2(x+5)-7 for x.

Solution:

1. Simplify:

4(x-2)+5=2(x+5)-7

4x-8+5=2x+10-7

4x-3=2x+3

2. Solve for the variables: we move the -3 to the right and the 2x to the left:

4x-3+3=2x+3+3

4x=2x+6

4x-2x=2x+6-2x

2x=6

3. Solve the equation: we divide by 2:

\frac{2}{2}x=\frac{6}{2}

x=3

EXAMPLE 5

  • Solve the equation x+14=4(x-5)+2(x+1)+7 for x.

Solution:

1. Simplify:

x+14=4(x-5)+2(x+1)+7

x+14=4x-20+2x+2+7

x+14=6x-11

2. Solve for the variables: we move the 14 to the right and the 6x to the left:

x+14-14=6x-11-14

x=6x-25

x-6x=6x-25-6x

-5x=-25

3. Solve the equation: we divide by -5:

\frac{-5}{-5}x=\frac{-25}{-5}

x=5


Linear equations – Practice problems

Solve the equation 2x+2=x+9.

Choose an answer






Solve the equation 3x+1=x-5.

Choose an answer






Find the value of x in the equation 5(2x-1)=3x+9.

Choose an answer






Find the value of x in the equation 2(x+3)=x+7.

Choose an answer






Solve the equation 3(t-1)=2(t+2).

Choose an answer







See also

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