College Algebra - Week 2

Linear Equations and Problem Solving

Linear equations are the engine of algebra. This week you extend equation-solving to multi-step problems, equations with variables on both sides, special cases (no solution and infinitely many solutions), and applied word problems. The skills here apply directly to every quantitative course you will take.

Learning Objectives

- Solve multi-step linear equations by combining like terms and using distribution. - Solve equations with variables on both sides. - Identify equations with no solution (contradiction) or infinitely many solutions (identity). - Solve literal equations for a specified variable. - Translate word problems into equations and solve. - Solve consecutive integer and age problems. - Solve basic geometry problems using perimeter and angle relationships.

1. Multi-Step Equations: The Strategy

Most linear equations require several steps. The general strategy: 1. Clear parentheses using the distributive property. 2. Combine like terms on each side. 3. Move variable terms to one side using addition or subtraction. 4. Move constant terms to the other side. 5. Divide (or multiply) to isolate the variable. 6. Check your answer. Following this order prevents most common errors.

2. Solving Multi-Step Equations

Example 1: Solve 3(x + 2) - 5 = 2x + 9. Distribute: 3x + 6 - 5 = 2x + 9 Combine: 3x + 1 = 2x + 9 Subtract 2x: x + 1 = 9 Subtract 1: x = 8 Check: 3(8 + 2) - 5 = 30 - 5 = 25. Right side: 2(8) + 9 = 25. Correct. Example 2: Solve 4(2x - 3) = 2(3x + 1). Expand: 8x - 12 = 6x + 2 Subtract 6x: 2x - 12 = 2 Add 12: 2x = 14 Divide: x = 7

3. Variables on Both Sides

When variables appear on both sides, collect them on one side first. Example: Solve 5x - 4 = 2x + 11. Subtract 2x from both sides: 3x - 4 = 11 Add 4: 3x = 15 Divide by 3: x = 5 Check: 5(5) - 4 = 21. Right: 2(5) + 11 = 21. Correct. It does not matter which side you collect variables on. Collecting on the side that gives a positive coefficient is usually more convenient.

4. No Solution and Infinitely Many Solutions

Sometimes all variables cancel. Case 1 — Contradiction (No Solution): Solve 3x + 5 = 3x - 2. Subtract 3x: 5 = -2. This is FALSE. The equation has no solution. The two sides describe parallel lines that never cross. Case 2 — Identity (Infinitely Many Solutions): Solve 2(x + 3) = 2x + 6. Expand: 2x + 6 = 2x + 6. Subtract 2x: 6 = 6. This is TRUE. Every real number is a solution. The two sides are always equal.

5. Literal Equations

A literal equation contains two or more variables. Solving for one variable means isolating it using the same steps as before. Example 1: Solve A = (1/2)bh for h. Multiply both sides by 2: 2A = bh Divide by b: h = 2A/b Example 2: Solve V = lwh for w. Divide by l: V/l = wh Divide by h: w = V/(lh) Example 3: Solve F = (9/5)C + 32 for C. Subtract 32: F - 32 = (9/5)C Multiply by 5/9: C = (5/9)(F - 32)

6. Translating Word Problems

Step 1: Identify the unknown. Assign it a variable. Step 2: Write an equation relating the unknown to the given information. Step 3: Solve the equation. Step 4: Answer the original question in context. Step 5: Check that the answer makes sense. Common phrase translations: - 'increased by' → + - 'less than' → subtracted from (be careful with order) - 'times as many' → multiplication - 'divided equally among' → division - 'is' → =

7. Consecutive Integer Problems

Consecutive integers differ by 1: n, n+1, n+2, ... Consecutive even or odd integers differ by 2: n, n+2, n+4, ... Example 1: The sum of two consecutive integers is 47. Find them. n + (n + 1) = 47 2n + 1 = 47 2n = 46 n = 23 Integers: 23 and 24. Example 2: The sum of three consecutive odd integers is 75. Find them. n + (n+2) + (n+4) = 75 3n + 6 = 75 3n = 69 n = 23 Integers: 23, 25, 27.

8. Age Problems

Age problems compare the current ages of two people or the same person at two different times. Example: Maria is 3 times as old as her daughter. In 8 years, she will be twice as old. Find their current ages. Let daughter's age = x. Maria's age = 3x. In 8 years: 3x + 8 = 2(x + 8) 3x + 8 = 2x + 16 x = 8 Daughter is 8, Maria is 24. Check in 8 years: daughter is 16, Maria is 32. Is 32 = 2*16? Yes.

9. Geometry: Perimeter and Angle Problems

Perimeter problems use perimeter formulas as the equation. Example 1: A rectangle has perimeter 54 cm. The length is 3 more than twice the width. Find the dimensions. Let width = w, length = 2w + 3. Perimeter: 2(w) + 2(2w + 3) = 54 2w + 4w + 6 = 54 6w = 48 w = 8, length = 19. Angle problems use the fact that angles in a triangle sum to 180. Example 2: Two angles of a triangle are equal. The third is 30 more than the equal angles. Find all angles. 2x + (x + 30) = 180 3x + 30 = 180 3x = 150 x = 50. Angles: 50°, 50°, 80°.

10. Coin and Mixture Problems

Coin problems count total value. Example: A jar has dimes and quarters totaling $3.35. There are 20 coins. How many of each? Let d = dimes, q = quarters. d + q = 20 → d = 20 - q 0.10d + 0.25q = 3.35 Substitute: 0.10(20 - q) + 0.25q = 3.35 2 - 0.10q + 0.25q = 3.35 0.15q = 1.35 q = 9 d = 11 Check: 11(0.10) + 9(0.25) = 1.10 + 2.25 = 3.35. Correct.

11. Distance-Rate-Time Problems

The key formula: distance = rate * time (d = rt). Example: Two cars leave from the same city at the same time, driving in opposite directions. One travels at 55 mph, the other at 65 mph. After how many hours are they 360 miles apart? Total distance = 55t + 65t = 360 120t = 360 t = 3 hours.

12. Common Mistakes

- Distributing only to the first term: 3(x + 4) is NOT 3x + 4. - Moving the wrong term: when you subtract 2x from 5x + 3 = 2x + 9, the result is 3x + 3 = 9, not 5x + 3 = 9. - Forgetting to flip the inequality (this matters in Week 3). - For 'less than' problems: '5 less than x' means x - 5, NOT 5 - x. - Declaring 'no solution' when a variable term cancels and a false statement results, but NOT when a true statement results (that is the infinitely-many-solutions case).