Let’s learn about linear equations.
What is a Linear Equation?
A linear equation is the equation for a straight line. Itβs 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 β 0, B β 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 β y1 = m(x β x1) | y β 3 = 6(x β 2) |
General form | Ax + By + C = 0 | 2x + 3y β 6 = 0 |
Whenever you make a change to an equation, you must do the EXACT SAME thing to BOTH sides of the equation. Hereβs a breakdown of rules in solving linear equations:
- 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.
- 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.
- 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.
- 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 β
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 β
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 β 0, b β 0; x and y are the variables.
Here are the methods for solving linear equations with two variables:
- Method of substitution
- Cross multiplication method
- Method of elimination
- 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 β ; -6 = -6 β
Thank you for reading. We hope itβs effective! Always feel free to revisit this page if you ever have any questions about linear equations.