Struggling with math variables? This step-by-step guide breaks down linear equations into simple, manageable steps. From basic properties of equality to complex examples with fractions, you'll find everything you need to master the basics and gain confidence in your algebra skills. Dive in to see solved examples and test yourself with our interactive quiz at the end!
An equation is an equality that contains a variable. You are required to find a number that, when substituted for the variable, yields a correct numerical equality (the same numbers on the left and right sides of the equality). In other words, you need to find the solution of the equation.
For example, in the equation 2x - 6x + 8 = 7x - 3, we can substitute 1 for the variable x and obtain a correct numerical equality, since 2(1) - 6(1) + 8 = 2 - 6 + 8 = 4 and 7(1) - 3 = 7 - 3 = 4. Therefore, x = 1 is a solution of the equation.
When solving equations, we may encounter the following cases: the equation has no solution,one solution, or infinitely many solutions.
In this post, we will look at how to solve equations that contain one variable to the first power. Such equations are called linear equations.
To solve such equations, you can apply
- The Subtraction property of equality. If a = b, then a - c =b - c;
- The Addition property of equality. If a = b, then a + c = b + c;
- The Division property of equality. If a = b, then \frac{a}{\color{Blue}c} = \frac{b}{\color{Blue}c};
- The Multiplication property of equality. If a = b, then ac = bc.
Examples
- 10x = 40
Divide both sides of the equation by 10: \frac{10x}{\color{Blue}10} = \frac{40}{\color{Blue}10}
Simplify. x = 4
Check: Substitute x = 4 10(4) = 40.
40 = 40. True. x = 4 is the solution - 8x + 9 = 25
Subtract 9 from both sides. 8x + 9 - 9= 25 - 9
Simplify. 8x = 16
Divide both sides of the equation by 8. \frac{8x}{\color{Blue}8} = \frac{16}{\color{Blue}8}
Simplify. x = 2
Check: Substitute x =2 8(2) + 9 = 25
16 + 9 = 25
25 = 25. True. x = 2 is the solution - 7x - 3x = 5
Combine like terms. 4x = 5
Divide both sides of the equation by 4: \frac{4x}{\color{Blue}4} = \frac{5}{\color{Blue}4}
Simplify. x = 1.25
Check: Substitute x =1.25 7(1.25) - 3(1.25) = 5
8.75 - 3.75 = 5
5 = 5. True. x = 1.25 is the solution - 3x - 2 = x + 6
3x - x = 6 + 2
Combine like terms. 2x = 8
Divide both sides of the equation by 2: \frac{2x}{\color{Blue}2} = \frac{8}{\color{Blue}2}
Simplify. x = 4
Check: Substitute x = 4 3(4) - 2 = 4 + 6
12 - 2 = 10
10 = 10. True. x = 4 is the solution - 3(2x - 4) = 5x + 1
Distribute on the left. 3⋅2x - 3⋅4 = 5x + 1
Simplify. 6x - 12 = 5x + 1
6x - 5x = 1 + 12
Simplify. x = 13
Check: Substitute x = 13 3(2(13) - 4) = 5(13) + 1
3(26 - 4) = 65 + 1
3(22) = 66
66=66. True. x = 13 is the solution - \frac{4}{9}x = \frac{8}{3}
Multiply by the reciprocal of \frac{4}{9}. {\color{Blue}\frac{9}{4}\cdot }\frac{4}{9}x ={\color{Blue}\frac{9}{4}\cdot } \frac{8}{3}
Multiply 1x = 2 ⋅ 3
Simplify. x = 6
Check: Substitute x = 6 \frac{4}{9}\cdot6 = \frac{8}{3}
\frac{24}{9} = \frac{8}{3}
\frac{8}{3} = \frac{8}{3}. True. x = 6 is the solution - \frac{x}{4} + \frac{x}{6} = 20
Multiply both sides of the equation by LCM (12): \frac{{\color{Blue}12}\cdot{x}}{4} + \frac{{\color{Blue}12}\cdot{x}}{6} = {\color{Blue}12}\cdot20
Simplify. 3x + 2x = 240
Combine like terms. 5x = 240
Divide both sides of the equation by 5: \frac{5x}{\color{Blue}5} = \frac{240}{\color{Blue}5}
Simplify. x = 48
Check: Substitute x = 48 \frac{48}{4} + \frac{48}{6} = 20
12 + 8 = 20
20 = 20. True. x = 48 is the solution
Check yourself by completing the interactive exercise
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