Linear Equations: Formula & Examples

Let’s learn about linear equations.

What is a Linear Equation?

A linear equation is the equation for a straight line. It’s a combination of constants and variables and is probably the simplest algebraic equations you’ll have to deal with.

The standard form of linear equation is expressed in the formula:
Ax + By + C = 0
where A, B, and C are constants; x and y are variables.
Also, A ≠ 0, B ≠ 0 and 1 is the highest exponent of variables.

How to Solve Linear Equations:

Before we jump right into solving linear equations, familiarize yourself first with several formulas for writing linear equations.

Linear Equation	General Form	Example
Slope intercept form	y = mx+b	y + 2x = 3
Point-slope form	y – y1 = m(x – x1)	y – 3 = 6(x – 2)
General form	Ax + By + C = 0	2x + 3y – 6 = 0

Whenever you make a change to an equation, you must do the EXACT SAME thing to BOTH sides of the equation. Here’s a breakdown of rules in solving linear equations:

1. If a=b then a+c = b+c for any c. You can add a number, c, to both sides of the equation and not change the equation.

2. If a=b then a-c = b-c for any c. You can subtract a number, c, from both sides of an equation and not change the equation.

3. If a=b then ac = bc for any c. Like addition and subtraction, you can multiply both sides of an equation by a number, c, and not change the equation.

4. If a=b then a/c = b/c for any non-zero c. You can divide both sides of an equation by a non-zero number, c, and not change the equation.

Solving Linear Equations with One Variable

Linear equations with one variable are expressed in the form of ax+b = 0, where a and b are two integers, and x is a variable with only one solution.

Example #1: One Variable

Find the value of x in the equation 5x = 30.

Step 1. Firstly, divide both sides by 5 to get the value of x.
5x = 30
5x / 5 = 30 / 5

Step 2. As a result of step 1, the 5 beside the x will be cancelled out.
5x / 5 = 30 /5
x = 6
Therefore, the value of x is-6.

Step 3. Check the solution by first substituting the value 6 for x into the original equation, and then performing the required operations.
5x = 30
5(6) = 30
30 = 30 ✓

Example #2: One Variable

Find the value of x in the equation 3x – 10 = 5x + 2.

Step 1. Combine like terms then solve the arithmetic operations.
3x – 10 = 5x +2
3x – 5x = 2 + 10
-2x = 12

Step 2. Then, divide both sides by -2. As a result, the -2 beside the x will be cancelled out.
-2x / -2 = 12 / -2
x = -6
Therefore, the value of x is -6.

Step 3. Check the solution by first substituting the value -6 for x into the original equation, and then performing the required operations.
3x – 10 = 5x +2
3(-6) – 10 = 5(-6) +2
-18 -10 = -30 + 2
-28 = -28 ✓

Solving Linear Equations with Two Variables

The standard form of a linear equation with two variables is represented as ax + by + c = 0
where, a ≠ 0, b ≠ 0; x and y are the variables.

Here are the methods for solving linear equations with two variables:

1. Method of substitution
2. Cross multiplication method
3. Method of elimination
4. Determinant methods

Example #3: Two Variables

Find the values of x and y in the equations below using the substitution method:
2x – y = 9 ; x + 3y = -6

Step 1. Firstly, look for the value of y. Use either equation to set the value of x. For example, let’s use x + 3y = -6. Therefore, we want to subtract 3y from both sides of the equation to get the value of x.
x + 3y – 3y = -6 – 3y
x = -6 -3y

Step 2. Since x = -6 -3y, substitute -6 – 3y for x in the other equation: 2x – y = 9
2x – y = 9
2(-6 – 3y) – y = 9

Step 3. Solve the resulting equation by first combining like terms and then performing the operations required. Lastly, simplify.
2(-6 -3y) – y = 9
-12 – 6y – y = 9
-6y – y = 9 + 12
-7y = 21
y = 21 / -7
y = -3
Therefore, the value of y is -3.

Step 4. Then, use the value of y to solve for x by substituting -3 for y in the equation:
x = -6 – 3y
x = -6 -3(-3) = -6 + 9 = 3
Therefore, the value of x is 3.

Finally, take the values of x and y to find the solution, which is (x, y) = (3, -3).

Related Reading: Interval Notation – Writing & Graphing

Step 5. Check the solution by first substituting 3 for x and -3 for y into the original equations, and then performing the required operations.
2x – y = 9 ; x + 3y = -6
2(3) – (-3) = 9 ; 3 + 3(-3) = -6
6 + 3 = 9 ; 3 – 9 = -6
9 = 9 ✓ ; -6 = -6 ✓

Thank you for reading. We hope it’s effective! Always feel free to revisit this page if you ever have any questions about linear equations.

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