Solving Linear Equations

A linear equation is an equation where the highest power of the variable is 1. In other words, the variable isn’t squared, cubed, or raised to any higher power. To solve a linear equation, you need to find the specific value of the variable that makes the equation correct or balanced.

The following steps can be used to solve a linear equation:

1. Simplify both sides of the equation, if necessary.

2. Collect all terms with the variable on one side of the equation and all constant terms on the other side.

3. Isolate the variable by dividing or multiplying both sides of the equation by the appropriate constant.

For example, if we want to solve the linear equation 4x - 7 = 2x + 5:

Step 1: Simplify both sides of the equation (if necessary) In this case, there’s no simplification needed on either side of the equation, so we proceed to the next step.

Step 2: Collect all terms with the variable on one side of the equation and all constant terms on the other side. To do this, we’ll subtract 2x from both sides of the equation to move all x terms to the left side.

4x - 2x - 7 = 2x - 2x + 5

This simplifies to:

2x - 7 = 5

Now, we’ll add 7 to both sides of the equation to move the constant term to the right side.

2x - 7 + 7 = 5 + 7

This simplifies to:

2x = 12

Step 3: Isolate the variable by dividing or multiplying both sides of the equation by the appropriate constant. In this case, we need to divide both sides by 2 to isolate x.

\frac{2x}{2} = \frac{12}{2}

This simplifies to:

x = 6

So, the solution to the linear equation 4x - 7 = 2x + 5 is x = 6.

Ok, let’s look at some more examples.

Examples

Example 1:

Solve the equation: 7x - 4 = 10

First, isolate the variable term by adding 4 to both sides of the equation.

7x - 4 + 4 = 10 + 47x - 4 + 4 = 10 + 4

7x = 147x = 14

Now, divide both sides by 7 to solve for x.

x = \frac{14}{7}x = \frac{14}{7}

x = 2x = 2

Example 2:

Solve the equation: 15x - 8 = 5x + 2

Begin by moving all x terms to one side and constants to the other side. In this case, subtract 5x from both sides and add 8 to both sides.

15x - 5x = 2 + 815x - 5x = 2 + 8

10x = 1010x = 10

Now, divide both sides by 10 to solve for x.

x = \frac{10}{10}x = \frac{10}{10}

x = 1x = 1

Example 3:

Solve the equation: \frac{x + 1}{3} = -2

First, eliminate the fraction by multiplying both sides by 3.

3(\frac{x + 1}{3}) = -2 \times 33(\frac{x + 1}{3}) = -2 \times 3

x + 1 = -6x + 1 = -6

Now, subtract 1 from both sides to solve for x.

x = -6 - 1x = -6 - 1

x = -7x = -7

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