Solving Linear Equations and Inequalities - Master High-Frequency Concepts and Skills for Algebra Proficiency—FAST

Easy Algebra Step-by-Step: Master High-Frequency Concepts and Skills for Algebra Proficiency—FAST! (2012)

Chapter 14. Solving Linear Equations and Inequalities

A linear equation in one variable, say x, has the standard form ax + b = c, a ≠ 0, where a, b, and c are constants. For example, 3x – 7 = 14 is a linear equation in standard form. An equation has two sides. The expression on the left side of the equal sign is the left side of the equation, and the expression on the right side of the equal sign is the right side of the equation.

Solving One-Variable Linear Equations

To solve a linear equation that has one variable x means to find a numerical value for x that makes the equation true. An equation is true when the left side has the same value as the right side. When you solve an equation, you undo what has been done to x until you get an expression like this: x = a number. As you proceed, you exploit the fact that addition and subtraction undo each other; and, similarly, multiplication and division undo one another.

The goal in solving a linear equation is to get the variable by itself on only one side of the equation and with a coefficient of 1 (usually understood).

You solve an equation using the properties of real numbers and simple algebraic tools. An equation is like a balance scale. To keep the equation in balance, when you do something to one side of the equation, you must do to the same thing to the other side of the equation.

Tools for Solving Linear Equations

Add the same number to both sides. Subtract the same number from both sides. Multiply both sides by the same nonzero number.

Divide both sides by the same nonzero number.

It is important to remember that when you are solving an equation, you must never multiply or divide both sides by 0.

Problem Solve the equation.

Solution

Step 1. The variable appears on both sides of the equation, so subtract 3x from the right side to remove it from that side. To maintain balance, subtract 3x from the left side, too.

Step 2. Simplify both sides by combining like variable terms.

Step 3. 9 is added to the variable term, so subtract 9 from both sides.

Step 4. Simplify both sides by combining constant terms.

Step 5. You want the coefficient of x to be 1, so divide both sides by 2.

Step 6. Simplify.

Step 7. Check your answer by substituting –5 for x in the original equation,

Substitute –5 for x on the left side of the equation: Similarly, on the right side, you have Both sides equal –16, so –5 is the correct answer.

Step 1. Use the distributive property to remove parentheses.

Step 2. 24 is subtracted from the variable term, so add 24 to both sides.

Step 3. Simplify both sides by combining constant terms.

4x = 64

Step 4. You want the coefficient of x to be 1, so divide both sides by 4.

Step 5. Simplify.

x = 16

Step 6. Check your answer by substituting 16 for x in the original equation, 4(x – 6) = 40.

Substitute 16 for x on the left side of the equation: On the right side, you have 40 as well. Both sides equal 40, so 16 is the correct answer.

Step 1. 7 is subtracted from the variable term, so add 7 to both sides.

Step 2. Simplify both sides by combining constant terms.

–3x = 21

Step 3. You want the coefficient of x to be 1, so divide both sides by –3.

Step 4. Simplify.

x = –7

Step 5. Check your answer by substituting –7 for x in the original equation, – 3x – 7 = 14.

Substitute –7 for x on the left side of the equation: –3x – 7 = –3(–7) – 7 = 21 – 7 = 14. On the right side, you have 14 as well. Both sides equal 14, so –7 is the correct answer.

Step 1. The variable appears on both sides of the equation, so add 2x to the right side to remove it from that side. To maintain balance, add 2x to the left side, too.

Step 2. Simplify both sides by combining like variable terms.

Step 3. 2 is subtracted from the variable term, so add 2 to both sides.

Step 4. Simplify both sides by combining constant terms.

5x = 9

Step 5. You want the coefficient of x to be 1, so divide both sides by 5.

Step 6. Simplify.

x = 1.8

Step 7. Check your answer by substituting 1.8 for x in the original equation,

Substitute 1.8 for x on the left side of the equation: Similarly, on the right side, you have Both sides equal 3.4, so 1.8 is the correct answer.

Step 1. Eliminate fractions by multiplying both sides by 10, the least common multiple of 2 and 5. Write 10 as to avoid errors.

Step 2. Simplify.

5(x – 3) = 2(2x + 4)

5x – 15 = 4x + 8

Step 3. The variable appears on both sides of the equation, so subtract 4x from the right side to remove it from that side. To maintain balance, subtract 4x from the left side, too.

Step 4. Simplify both sides by combining variable terms.

Step 5. 15 is subtracted from the variable term, so add 15 to both sides.

Step 6. Simplify both sides by combining constant terms.

x = 23

Step 7. Check your answer by substituting 23 for x in the original equation, .

Substitute 23 for x on the left side of the equation:

. Similarly, on the right side, you have .

. Both sides equal 10, so 23 is the correct answer.

Solving Two-Variable Linear Equations for a Specific Variable

You can use the procedures for solving a linear equation in one variable x to solve a two-variable linear equation, such as 6x + 2y = 10, for one of the variables in terms of the other variable. As you solve for the variable of interest, you simply treat the other variable as you would a constant. Often, you need to solve for y to facilitate the graphing of an equation. (See Chapter 17 for a fuller discussion of this topic.) Here is an example.

Problem Solve 6x + 2y = 10 for y.

Solution

Step 1. 6x is added to the variable term 2y, so subtract 6x from both sides.

When you are solving 6x 2y = 10 for y, treat 6x as if it were a constant.

Step 2. Simplify.

Step 3. You want the coefficient of y to be 1, so divide both sides by 2.

You must divide both terms of the numerator by 2.

Step 4. Simplify.

y = 5 – 3x

Solving Linear Inequalities

If you replace the equal sign in a linear equation with <, >, ≤, or ≥, the result is a linear inequality. You solve linear inequalities just about the same way you solve equations. There is just one important difference. When you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality symbol. To help you understand why you must do this, consider the two numbers, 8 and 2. You know that 8 > 2 is a true inequality because 8 is to the right of 2 on the number line, as shown in Figure 14.1.

Figure 14.1 The numbers 2 and 8 on the number line

If you multiply both sides of the inequality 8 > 2 by a negative number, say, –1, you must reverse the direction of the inequality so that you will still have a true inequality, namely, –8 < –2. You can verify that –8 < –2 is a true inequality by observing that –2 is to the right of –8 on the number line as shown in Figure 14.2.

Figure 14.2 The numbers –8 and –2 on the number line

If you neglect to reverse the direction of the inequality symbol after multiplying both sides of 8 > 2 by –1, you get the false inequality –8 > –2.

Problem Solve the inequality.

Solution

Step 1. The variable appears on both sides of the inequality, so subtract 3x from the right side to remove it from that side. To maintain balance, subtract 3x from the left side, too.

Step 2. Simplify both sides by combining like variable terms.

Step 3. 6 is added to the variable term, so subtract 6 from both sides.

When solving an inequality, do not reverse the direction of the inequality symbol because of subtracting the same number from both sides.

Step 4. Simplify both sides by combining constant terms.

Step 5. You want the coefficient of x to be 1, so divide both sides by 2.

When solving an inequality, do not reverse the direction of the inequality symbol because of dividing both sides by a positive number.

Step 6. Simplify.

x < –4 is the answer.

Step 1. Use the distributive property to remove parentheses.

Step 2. 24 is subtracted from the variable term, so add 24 to both sides.

When solving an inequality, do not reverse the direction of the inequality because of adding the same number to both sides.

Step 3. Simplify both sides by combining constant terms.

Step 4. You want the coefficient of x to be 1, so divide both sides by 4.

Step 5. Simplify.

x ≥ 17 is the answer.

Step 1. 7 is subtracted from the variable term, so add 7 to both sides.

Step 2. Simplify both sides by combining constant terms.

–3x > 21

Step 3. You want the coefficient of x to be 1, so divide both sides by –3 and reverse the direction of the inequality because you divided by a negative number.