Create Equations: One Variable Context

Grade 9 · algebra · 102 practice problems · read aloud

🔊 Listen to this explanation

Creating Equations in Algebra

What is Creating Equations? 🤔

Creating an equation means translating a real-world situation or word problem into a mathematical sentence using variables. This is useful because it lets you solve complex problems from science, business, and everyday life. For example, you can figure out profits, distances, or how much material you need for a project.

Step-by-Step Guide

  1. Step 1: Understand the Problem. Read carefully. What are you trying to find?
  2. Step 2: Define a Variable. Choose a letter (like x) to represent the unknown quantity.
  3. Step 3: Translate Words into Math. Look for key phrases: "sum" means +, "difference" means -, "product" means ×, "quotient" means ÷.
  4. Step 4: Write the Equation. Assemble the parts into an equation using your variable.
  5. Step 5: Solve and Check. Solve for the variable. Does your answer make sense in the original problem?

Visual Examples

Example 1: The Concert

Ticket for a concert cost $25. The total money collected was $1800. How many tickets were sold?

  1. Define: Let t = number of tickets sold.
  2. Translate: (cost per ticket) × (number of tickets) = total money. So, 25 × t = 1800.
  3. Equation: 25t = 1800
  4. Solve: t = 1800 / 25 = 72.
  5. Check: 25 × 72 = 1800. ✔️ 72 tickets were sold.

Example 2: Consecutive Integers

The sum of three consecutive integers is 48. Find the integers.

  1. Define: Let n = the first integer. Then n+1 = the second, and n+2 = the third.
  2. Translate: (first) + (second) + (third) = 48.
  3. Equation: n + (n+1) + (n+2) = 48
  4. Solve: 3n + 3 = 48 → 3n = 45 → n = 15.
  5. Check: 15 + 16 + 17 = 48. ✔️ The integers are 15, 16, and 17.

Common Mistakes to Avoid

Misplacing the Equals Sign: The equals sign (=) means "is the same as." Everything on the left must balance with everything on the right.

Ignoring the Order of Operations: Remember PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction) when writing or solving.

Incorrect Variable Definition: Be clear about what your variable represents. Write it down: "Let x = ..."

Tips & Tricks

Key Word Translation: "More than" often means addition, but the order can be tricky. "5 more than x" is x + 5, not 5 + x.

Draw a Diagram: For problems involving geometry or movement, a quick sketch can make the equation obvious.

Isolate the Variable: To solve, use inverse operations. If it's multiplied, divide. If it's added, subtract. Do it to both sides!

How to Practice

  • Start with simple word problems from your textbook and focus purely on writing the equation, not solving it.
  • Create your own word problems for a given equation, like 2x + 10 = 30.
  • Work with a friend and "think aloud," explaining your reasoning for each step as you define the variable and build the equation.

Practice problems

6 of the 102, worked through step by step — try them before opening the answer.

1 2x² - 8x + 8 = 0

Hint: Look for a common factor first, then solve the simplified quadratic equation

Show the answer

Answer: 2

  1. Factor out the common factor of 2 from all terms: 2(x² - 4x + 4) = 0
  2. Divide both sides by 2: x² - 4x + 4 = 0
  3. Recognize this as a perfect square trinomial: (x - 2)² = 0
  4. Take the square root of both sides: x - 2 = 0
  5. Add 2 to both sides: x = 2

The answer is 2.

2 3x² - 12x + 12 = 0

Hint: This is a quadratic equation that can be simplified by factoring out a common factor first. Look for a perfect square trinomial pattern after simplification.

Show the answer

Answer: 2

  1. Factor out the greatest common factor, which is 3 3(x² - 4x + 4) = 0
  2. Divide both sides by 3 x² - 4x + 4 = 0
  3. Recognize this as a perfect square trinomial (x - 2)² = 0
  4. Take the square root of both sides x - 2 = 0
  5. Solve for x x = 2

The answer is 2.

3 Aroha is designing a rectangular banner with perimeter 120 cm. The length is 4x + 6 cm and the width is 2x - 4 cm. Write an equation for x.

Hint: Remember that perimeter is calculated by adding all sides of a rectangle. The formula for perimeter of a rectangle is 2 times length plus 2 times width.

Show the answer

Answer: 2(4x + 6) + 2(2x - 4) = 120

  1. Write the perimeter formula for a rectangle: P = 2L + 2W
  2. Substitute the given values: P = 120, L = 4x + 6, W = 2x - 4
  3. Plug into the formula: 2(4x + 6) + 2(2x - 4) = 120
  4. This is the equation that can be solved for x
  5. To verify, we can simplify: 8x + 12 + 4x - 8 = 120 → 12x + 4 = 120 → 12x = 116 → x = 116/12 = 29/3 The equation is 2(4x + 6) + 2(2x - 4) = 120

4 Aroha has a rectangular garden with an area of 63 square meters. The length is 7 meters more than the width. Write an equation to find the width.

Hint: Remember that area of a rectangle is length times width. If width is the variable, express length in terms of that variable.

Show the answer

Answer: w(w + 7) = 63

  1. Let w represent the width of the garden.
  2. Since the length is 7 meters more than the width, the length is w + 7.
  3. The area of a rectangle is length × width, and we know the area is 63 square meters.
  4. Write the equation: w(w + 7) = 63
  5. This equation can be used to solve for the width.

5 Emma is designing a rectangular banner. The length is (2x + 5) cm and the width is x cm. If the perimeter is 150 cm, write an equation to find x.

Hint: The perimeter of a rectangle is found by adding all four sides. Think about how to express the total distance around using the given length and width expressions.

Show the answer

Answer: 2(2x + 5 + x) = 150

  1. The formula for the perimeter of a rectangle is P = 2(length + width).
  2. Substitute the given expressions: length = (2x + 5) cm, width = x cm, and perimeter = 150 cm.
  3. This gives us the equation: 2((2x + 5) + x) = 150.
  4. Simplify inside the parentheses: 2(3x + 5) = 150. The equation to find x is 2(3x + 5) = 150.

6 Aroha has a rectangular garden with perimeter 72 meters. The length is 2x+4 meters and the width is x meters. Write an equation for the perimeter.

Hint: Remember that perimeter of a rectangle is calculated by adding all four sides together. Think about how to express the total perimeter using the given expressions for length and width.

Show the answer

Answer: 2(2x+4)+2x=72

  1. The perimeter of a rectangle is given by P = 2(length) + 2(width)
  2. Substitute the given expressions: length = 2x+4, width = x, perimeter = 72
  3. Write the equation: 2(2x+4) + 2(x) = 72
  4. This is the required equation: 2(2x+4) + 2x = 72
Practise this topic — 10 free problems, no signup →