Lesson 4.1: Solving Linear Equations

Introduction

In this lesson, we will explore the fundamentals of solving linear equations, an essential concept in algebra. Linear equations form the basis for many applications in mathematics and everyday life. By the end of this lesson, you will be able to solve one-step and two-step linear equations, handle equations with unknowns on both sides, and solve equations that incorporate brackets and fractions.

Learning Objectives

• Solve one-step and two-step linear equations.
• Tackle equations with the unknown on both sides, with brackets, and with fractions.
• Check a solution by substitution.
• Solve linear equations with the unknown on both sides.
• Solve linear equations involving brackets and fractions.

What is a Linear Equation?

A linear equation is an equation of the first degree, meaning that it involves variables raised only to the first power. The general form of a linear equation in one variable is:

$$

and + b = 0$$

where $a$ and $b$ are constants, and $x$ is the variable. The solutions to linear equations can be visualized as points on a number line.

Solving One-Step Linear Equations

One-step linear equations are the simplest form of equations, requiring only one operation to solve for the variable. Let’s take a look at the steps involved in solving a one-step linear equation.

Example 1: Solve $x + 5 = 12$

1. Identify the equation: We have $x + 5 = 12$.
2. Isolate the variable: Subtract 5 from both sides:

$$x + 5 - 5 = 12 - 5$$

This simplifies to:

$$x = 7$$

1. Check your solution: Substitute $x$ back in the original equation:

$$7 + 5 = 12$$

Since this is true, our solution is correct.

Common Misconception

A common mistake is to forget to perform the same operation on both sides of the equation. Always remember that whatever you do to one side must also be done to the other side to maintain the equality.

Solving Two-Step Linear Equations

Two-step equations require two operations to isolate the variable. Let’s explore how to solve these types of equations.

Example 2: Solve $2x + 3 = 11$

1. Identify the equation: Our equation is $2x + 3 = 11$.
2. Isolate the variable: First, subtract 3 from both sides:

$$2x + 3 - 3 = 11 - 3$$

This simplifies to:

$$2x = 8$$

1. Divide both sides by 2:

$$x = \frac{8}{2}$$

Which simplifies to:

$$x = 4$$

1. Check your solution: Substitute back into the original equation:

$$2(4) + 3 = 11$$

Since this is true, our solution is confirmed.

Linear Equations with Unknowns on Both Sides

Sometimes, we will encounter equations with variables on both sides. We will need to rearrange them to isolate the variable.

Example 3: Solve $3x + 7 = 2x + 12$

1. Identify the equation: The equation is $3x + 7 = 2x + 12$.
2. Rearrange the equation: Subtract $2x$ from both sides:

$$3x - 2x + 7 = 12$$

This simplifies to:

$$x + 7 = 12$$

1. Isolate the variable: Subtract 7 from both sides:

$$x = 12 - 7$$

Thus, we get:

$$x = 5$$

1. Check your solution: Substitute back:

$$3(5) + 7 = 2(5) + 12$$

Since $15 + 7 = 10 + 12$, which simplifies to $22 = 22$, our solution checks out.

Linear Equations Involving Brackets

Linear equations can include brackets that we need to expand before solving. Let's see how to handle those.

Example 4: Solve $2(x + 4) = 16$

1. Expand the brackets: Multiply out:

$$2x + 8 = 16$$

1. Isolate the variable: Subtract 8 from both sides:

$$2x = 16 - 8$$

This simplifies to:

$$2x = 8$$

1. Divide both sides by 2:

$$x = \frac{8}{2}$$

Which simplifies to:

$$x = 4$$

1. Check your solution: Substitute back:

$$2(4 + 4) = 16$$

Since this is true, our solution works.

Linear Equations with Fractions

When solving linear equations with fractions, it's often helpful to eliminate the fractions first by multiplying through by the least common denominator (LCD).

Example 5: Solve $\frac{1}{2}x + 3 = 5$

1. Identify the equation: We have $\frac{1}{2}x + 3 = 5$.
2. Eliminate fractions: The LCD here is 2. Multiply each term by 2:

$$2 \times \left(\frac{1}{2}x

ight) + $2 \times 3$ = $2 \times 5$$$

This simplifies to:

$$x + 6 = 10$$

1. Isolate the variable: Subtract 6 from both sides:

$$x = 10 - 6$$

Thus, we get:

$$x = 4$$

1. Check your solution: Substitute back:

$$\frac{1}{2}(4) + 3 = 5$$

Since $2 + 3 = 5$, our solution is confirmed.

Conclusion

In this lesson, we covered the fundamental concepts of solving linear equations, including one-step and two-step equations, those with unknowns on both sides, brackets, and fractions. You have learned how to isolate variables using inverse operations and the importance of maintaining equality by performing the same operations on both sides of the equation. Mastering these techniques will empower you to tackle more complex algebraic problems in the future.

Study Notes

• A linear equation is an equation involving variables raised to the first power.
• One-step equations require one operation to isolate the variable.
• Two-step equations require two operations to solve.
• To solve equations with unknowns on both sides, rearrange to isolate the variable.
• Expand brackets before solving equations that involve them.
• Eliminate fractions by multiplying through by the least common denominator.
• Always check your solutions by substituting them back into the original equation.