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Simplifying Square Roots
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Adding and Subtracting Fractions
Solving Linear Equations
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Recognizing Polynomial Equations from their Graphs
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Factoring a Sum or Difference of Two Cubes
Solving Nonlinear Equations by Substitution
Solving Systems of Linear Inequalities
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Finding the Equation of an Inverse Function
Plotting Points in the Coordinate Plane
The Product of the Roots of a Quadratic
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Solving Quadratic Equations by Completing the Square
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Solving Linear Equations

Strategy for Solving Linear Equations

The most basic equations of algebra are linear equations. There is even a connection between linear equations in one variable and straight lines.

 

Linear Equation in One Variable

A linear equation in one variable x is an equation of the form ax + b = 0, where a and b are real numbers, with a ≠ 0.

A linear equation has exactly one solution. The strategy that we use for solving linear equations is summarized in the following lines.

 

Strategy for Solving a Linear Equation

1. If fractions are present, multiply each side by the LCD to eliminate them.

2. Use the distributive property to remove parentheses.

3. Combine any like terms.

4. Use the addition property of equality to get all variables on one side and numbers on the other side.

5. Use the multiplication property of equality to get a single variable on one side.

6. Check by replacing the variable in the original equation with your solution.

Note that not all equations require all of the steps.

 

Example 1

Using the equation-solving strategy

Solve the equation

Solution

We first multiply each side of the equation by 10, the LCD for 2, 5, and 10. However, we do not have to write down that step. We can simply use the distributive property to multiply each term of the equation by 10.

 
Multiply each side by 10.
5y - 2(y - 4) = 23 Divide each denominator into 10 to eliminate fractions.
5y - 2y + 8 = 23 Be careful to change all signs: -2(y - 4) = -2y + 8
3y + 8 = 23 Combine like terms.
3y + 8 - 8 = 23 - 8 Subtract 8 from each side.
3y = 15 Simplify
Divide each side by 3.
y = 5  

Check that 5 satisfies the original equation. The solution set is {5}.

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