Section 1: Solving Linear Equations
It is important that you watch the video first.
A linear equation is an equation which contains a variable like "x," rather than something like x2. Linear equations may look like x + 6 = 4, or like 2a – 3 = 7.
In general, in order to solve an equation, you want to get the variable by itself by undoing any operations that are being applied to it.
Here is a general strategy to use when solving linear equations.
Solving Linear Equations
- Clear fractions or decimals.
- Simplify each side of the equation by removing parentheses and combining like terms.
- Isolate the variable term on one side of the equation.
- Solve the equation by dividing each side of the equation.
- Check your solution.
Example 1: Solve for x: 3(2 – 5x) + 4(6x) = 12
Solution.
| Step 1. | Clear fractions or decimals. |
|---|---|
| This step is not necessary for the given equation. |
|
| Step 2 | Simplify each side of the equation. |
| Remove parentheses | 3(2 – 5x) + 4(6x) = 12 |
| Apply the distributive property. | 6 – 15x + 24x = 12 |
| Combine like terms | 6 – 15x + 24x = 12 |
| The x-terms combine on the left side of the equation. | 6 + 9x = 12 |
| Step 3. | Isolate the variable term on one side of the equation. |
| Subtract 6 from each side of the equation. | 6 + 9x = 12 6 + 9x – 6 = 12–6 9x = 6 |
| Step 4. | Solve the equation by dividing each side of the equation. |
| Divide each side of the equation by 9. | 9x ÷ 9 = 6 ÷ 9 |
| Reduce the fraction. | x = 2/3 |
| Step 5. Check your solution. | This is left up to you to do. |
Example 2: Solve for y: 0.12(y – 6) + 0.06y = 0.08y – 0.7
Solution.
| Step 1 | Clear fractions or decimals. |
|---|---|
| Multiply each side of the equation by 100. |
100[0.12(y – 6) + 0.06y ] =100[0.08y – 0.7]
|
| Step 2. | Simplify each side of the equation. |
| Distribute the 100 to each term of the equation. |
100[0.12(y – 6) ] + 100[0.06y ] =100[0.08y] – 100[0.7]
|
| Simplify terms | 12(y – 6) + 6y = 8y – 70 |
| Remove parentheses | 12y – 72 + 6y = 8y – 70 |
| Combine like terms | 18y – 72 = 8y – 70 |
| Step 3. | Isolate the variable term on one side of the equation. |
| Subtract 8y from each side of the equation. | 18y – 72 – 8y = 8y – 70 – 8y 10y – 72 = – 70 |
| Add 72 to each side of the equation. | 10y – 72 + 72 = – 70 + 72 10y = 2 |
| Step 4. | Solve the equation by dividing each side of the equation. |
|
Divide each side of the equation by 10. Reduce the fraction. |
10y ÷ 10 = 2 ÷ 10 y = 1/5 = 0.2 |
| Step 5. Check your solution | This is left up to you to do |
Solving Linear Equations which either have No Solution
Example 3: Solve the following equation by factoring.
Solve for x: 2(x + 3) – 5 = 5x – 3(1 + x)
Solution
|
Step 1. Clear fractions or decimals. | |
|---|---|
| This step is not necessary for the given equation | 2(x + 3) – 5 = 5x – 3(1 + x) |
| Step 2. | Simplify each side of the equation. |
|
Remove parentheses Combine like terms |
2x + 6 – 5 = 5x – 3 – 3x 2x + 6 – 5 = 5x – 3 – 3x 2x + 1 = 2x – 3 |
| Step 3. | Isolate the variable term on one side of the equation. |
| Subtract 2x from each side of the equation | 2x + 1 – 2x = 2x – 3 – 2x 1 = – 3 |
| Since the final equation contains no variable terms, and the equation that is left is a false equation, there is no solution to this equation. The equation is also called a contradiction. | |
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