2 Equations, Inequalities, and Applications

2.3 More on Solving Linear Equations Objectives 1. Learn and use the four steps for solving a linear equation. 2. Solve equations that have no solution or infinitely many solutions. 3. Solve equations with fractions or decimals as coefficients. 4. Write expressions for two related unknown quantities.

Solving a Linear Equation Step 1 Simplify each side separately. Clear (eliminate) parentheses, fractions, and decimals, using the distributive property as needed, and combine like terms. Step 2 Isolate the variable term on one side. Use the addition property so that the variable term is on one side of the equation and a number is on the other. Step 3 Isolate the variable. Use the multiplication property to get the equation in the form x = a number, or a number = x. (Other letters may be used for the variable.) Step 4 Check. Substitute the proposed solution into the original equation to see if a true statement results. If not, rework the problem.

Using Both Properties of Equality Example 1 Solve the equation. No Step 1 –6x + 5 = 17 Step 2 –6x + 5 – 5 = 17 – 5 Subtract 5. –6x = 12 Combine terms. Divide by –6. Step 3 w = –2 Step 4

Using Both Properties of Equality Example 2 Solve the equation. Step 1 5w + 3 – 2w – 7 = 6w + 8 3w – 4 = 6w + 8 Combine terms. Step 2 3w – 4 + 4 = 6w + 8 + 4 Add 4. 3w = 6w + 12 Combine terms. 3w – 6w = 6w + 12 – 6w Subtract 6w. –3w = 12 Combine terms. Step 3 Divide by –3. w = –4

Using Both Properties of Equality Example 2 (continued) Solve the equation. Step 4 Check by substituting –4 for w in the original equation. 5w + 3 – 2w – 7 = 6w + 8 5(–4) + 3 – 2(–4) – 7 = 6(–4) + 8 ? Let w = –4. –20 + 3 + 8 – 7 = –24 + 8 ? Multiply. –16 = –16 True The solution to the equation is –4.

Solving a Linear Equation Example 3 Solve the equation. 5 ( h – 4 ) + 2 = 3h – 4 5h – 20 + 2 = 3h – 4 Distribute. 5h – 18 = 3h – 4 Combine terms. 5h – 18 + 18 = 3h – 4 + 18 Add 18. 5h = 3h + 14 Combine terms. 5h – 3h = 3h + 14 – 3h Subtract 3h. 2h = 14 Combine terms. Divide by 2. h = 7

Solving a Linear Equation Example 3 (continued) Solve the equation. Step 4 Check by substituting 7 for h in the original equation. 5 ( h – 4 ) + 2 = 3h – 4 5 ( 7 – 4 ) + 2 = 3(7) – 4 ? Let h = 7. 5 (3) + 2 = 3(7) – 4 ? Subtract. 15 + 2 = 21 – 4 ? Multiply. 17 = 17 True The solution to the equation is 7.

Solving a Linear Equation Example 5 Solve the equation. Step 1 15y – ( 10y – 2 ) = 2 ( 5y + 7 ) – 16 1 15y – 10y + 2 = 10y + 14 – 16 Distribute. 5y + 2 = 10y – 2 Combine terms. 5y + 2 – 2 = 10y – 2 – 2 Subtract 2. Step 2 5y = 10y – 4 Combine terms. 5y – 10y = 10y – 4 – 10y Subtract 10y. –5y = – 4 Combine terms. Divide by –5. Step 3

Solving a Linear Equation Example 5 (continued) Solve the equation. Check by substituting for y in the original equation. 4 5 Step 4 15y – ( 10y – 2 ) = 2 ( 5y + 7 ) – 16 Let y = . 4 5 15 – ( 10 – 2 ) = 2 ( 5 + 7 ) – 16 4 5 12 – ( 8 – 2 ) = 2 ( 4 + 7 ) – 16 Multiply. 12 – 6 = 2 ( 11 ) – 16 Subtract, add. 12 – 6 = 22 – 16 Multiply. 6 = 6 True The solution to the equation is

Solving an Equation That Has Infinitely Many Solutions Example 6 Solve the equation. 4 ( 2n + 6 ) = 2 ( 3n + 12 ) + 2n 8n + 24 = 6n + 24 + 2n Distribute. 8n + 24 = 8n + 24 Combine terms. 8n + 24 – 24 = 8n + 24 – 24 Subtract 24. 8n = 8n Combine terms. 8n – 8n = 8n – 8n Subtract 8n. 0 = 0 True Solution Set: {all real numbers}.

Solve Equations with No Solution or Infinitely Many Solutions An equation with both sides exactly the same, like 0 = 0, is called an identity. An identity is true for all replacements of the variables. We write the solution set as {all real numbers}. CAUTION In example 6, do not write {0} as the solution set of the equation. While 0 is a solution, there are infinitely many other solutions. 12

Solving an Equation That Has No Solution Example 7 Solve the equation. 6x – ( 4 – 3x ) = 8 + 3 ( 3x – 9 ) 1 6x – 4 + 3x = 8 + 9x – 27 Distribute. 9x – 4 = –19 + 9x Combine terms. 9x – 4 – 9x = –19 + 9x – 9x Subtract 9x. –4 = –19 False There is no solution. Solution set:

Solve Equations with No or Infinitely Many Solutions Again, the variable has disappeared, but this time a false statement (– 4 = – 19) results. This is the signal that the equation, called a contradiction, has no solution. Its solution set is the empty set, or null set, symbolized . CAUTION Do not write { } to represent the empty set. 14

Solve Equations with Fraction or Decimal Coefficients We clear an equation of fractions by multiplying each side by the least common denominator (LCD) of all the fractions in the equation. It is a good idea to do this in Step 1 to avoid messy computations. When clearing an equation containing decimals, choose the smallest exponent on 10 needed to eliminate the decimals.

Solving an Equation with Fractions as Coefficients Example 8 Solve the equation. 5 8 m 3 4 1 2 – 10 = + 5 8 m 3 4 1 2 – 10 = + Multiply by LCD, 8. 5 8 m 3 4 1 2 – 10 = + Distribute. 5m – 80 = 6m + 4m Multiply. Now use the four steps to solve this equivalent equation.

Solving an Equation with Fractions as Coefficients Example 8 (continued) Solve the equation. 5m – 80 = 6m + 4m Step 1 5m – 80 = 10m Combine terms. Step 2 5m – 80 – 5m = 10m – 5m Subtract 5m. – 80 = 5m Combine terms. Step 3 – 80 5m = 5 5 Divide by 5. – 16 = m

Solving an Equation with Fractions as Coefficients Example 8 (continued) Solve the equation. Step 4 Check by substituting –16 for m in the original equation. 5 8 m 3 4 1 2 – 10 = + 5 8 (–16) 3 4 1 2 – 10 = + ? Let m = –16. –10 – 10 = –12 – 8 ? Multiply. –20 = –20 True The solution to the equation is –16.

Solve Equations with Fraction or Decimal Coefficients Note Multiplying by 10 is the same as moving the decimal point one place to the right. Example: 1.5 ( 10 ) = 15. Multiplying by 100 is the same as moving the decimal point two places to the right. Example: 5.24 ( 100 ) = 524. Multiplying by 10,000 is the same as moving the decimal point how many places to the right? Answer: 4 places.

Solving an Equation with Decimals as Coefficients Example 10 Solve the equation. 0.2v – 0.03 ( 11 + v ) = – 0.06 ( 31 ) 20v – 3 ( 11 + v ) = – 6 ( 31 ) Multiply by 100. Step 1 20v – 33 – 3v = – 186 Distribute. 17v – 33 = – 186 Combine terms.

Solving an Equation with Decimals as Coefficients Example 10 (continued) Solve the equation. 17v – 33 = – 186 From Step 1 Step 2 17v – 33 + 33 = – 186 + 33 Add 33. 17v = – 153 Combine terms. 17 17 17v – 153 = Step 3 Divide by 17. Check to confirm that – 9 is the solution. v = – 9 21

Write Expressions for Two Related Unknown Quantities Example 11a Two numbers have a sum of 32. If one of the numbers is represented by c, write an expression for the other number. Given: c represents one number. The sum of the two numbers is 32. Solution: 32 – c represents the other number. Check: One number + the other number = 32 c + (32 – c) = 32

Writing Expressions for Two Related Unknown Quantities Example 11b Two numbers have a product of 24. If one of the numbers is represented by x, find an expression for the other number. x represents one of the numbers To find the other number, we would divide .