Infinite Solutions

Grade 8 · algebra · 66 practice problems · read aloud

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Infinite Solutions in Algebra 🔁

When solving equations, sometimes you end up with a statement that is always true, like 5 = 5. This means that no matter what value you plug in for the variable, the equation will hold true. Therefore, there are infinite solutions—an unlimited number of answers that work!

How to Identify Infinite Solutions

  1. Simplify Both Sides: Use the distributive property and combine like terms on each side of the equation.
  2. Get Variables on One Side: Add or subtract terms to move all variable terms to one side of the equals sign.
  3. Analyze the Result:
    • If you get a true statement without a variable (e.g., 3 = 3), you have infinite solutions.
    • If you get a false statement (e.g., 2 = 5), you have no solution.

Worked Examples

Example 1: 2(x + 3) = 2x + 6

  1. Distribute: 2x + 6 = 2x + 6
  2. Subtract 2x from both sides: 6 = 6 ✅
  3. This is a TRUE statement. Therefore, there are infinite solutions.

Example 2: 4n + 2 = 2(2n + 1)

  1. Distribute on the right: 4n + 2 = 4n + 2
  2. Subtract 4n from both sides: 2 = 2 ✅
  3. This is also a TRUE statement. Infinite solutions!

⚠️ Common Mistakes

  • Stopping Too Soon: Don't stop when both sides look similar. You must simplify until the variable disappears and you get a true statement.
  • Confusing with "No Solution": Remember, a true statement (2=2) means infinite solutions. A false statement (2=5) means no solution.
  • Forgetting the Distributive Property: Always distribute coefficients before moving terms. For example, in 3(x + 2) = 3x + 5, you must distribute to get 3x + 6 = 3x + 5, which leads to 6=5 (false), meaning no solution.

💡 Tips & Tricks

  • The Twin Test: If the two sides of the equation are identical twins after simplifying, you have infinite solutions.
  • Quick Check: Pick any number for the variable and plug it in. If it works for two different numbers you try, you've likely got infinite solutions.
  • Memory Aid: "True Statement, Infinite Wonder. False Statement, No Answer."

How to Practice

To master this concept, try these activities:

  • Create your own equations with infinite solutions by writing the same expression on both sides (e.g., 5x - 1 = 5x - 1).
  • Practice with online worksheets that mix equations with one, none, and infinite solutions.
  • Work with a partner. Each person solves an equation and explains why it has one, none, or infinite solutions.

Practice problems

6 of the 66, worked through step by step — try them before opening the answer.

1 6(2x - 4) + 24 = 12x ?

Hint: Simplify both sides of the equation by distributing and combining like terms. Then compare the final expressions on each side. If they are identical, what does that tell you about the number of solutions?

Show the answer

Answer: infinitely many solutions

  1. Distribute the 6 on the left side: 6(2x - 4) = 12x - 24
  2. The equation becomes: 12x - 24 + 24 = 12x
  3. Combine like terms on the left: -24 + 24 = 0, so we have 12x = 12x
  4. Subtract 12x from both sides: 12x - 12x = 12x - 12x, which gives 0 = 0
  5. Since 0 = 0 is always true for any value of x, the original equation is an identity. This means every real number is a solution.

The answer is infinitely many solutions.

2 4(3x - 2) + 8 = 12x = ?

Hint: Simplify both sides of the equation by distributing and combining like terms. Look for what value makes the equation true for all x.

Show the answer

Answer: 0

  1. Distribute the 4 on the left side: 4(3x - 2) = 12x - 8
  2. The equation becomes: 12x - 8 + 8 = 12x
  3. Combine like terms: -8 + 8 = 0, so we have 12x = 12x
  4. Subtract 12x from both sides: 12x - 12x = 12x - 12x
  5. This gives us 0 = 0
  6. Since this is always true regardless of x, the equation has infinitely many solutions. The question asks for the value of the expression 12x = ?, but since the equation is an identity, we need to find what makes it true for all x. Looking at the original equation 4(3x - 2) + 8 = 12x, when we simplify we get 12x = 12x, which means both sides are equal for any value of x. The question is asking for the value of the expression 12x = ?, but in the context of the equation being an identity,

the answer is 0 because when we rearrange we get 0 = 0. The answer is 0.

3 3(2x + 4) - 2(3x + 6) = ?

Hint: When simplifying expressions, distribute the coefficients first, then combine like terms. For example, with 2(a + b) - 2(a + c), distribute then subtract.

Show the answer

Answer: 0

  1. Distribute the 3: 3(2x + 4) = 6x + 12
  2. Distribute the -2: -2(3x + 6) = -6x - 12
  3. Combine the expressions: 6x + 12 - 6x - 12
  4. Combine like terms: (6x - 6x) + (12 - 12) = 0x + 0
  5. Simplify: 0x + 0 = 0

The answer is 0.

4 Liam is designing a rectangular garden with a length that is 3 meters more than twice its width. The perimeter of the garden is 36 meters. What are the dimensions of Liam's garden?

Hint: Try defining the width as a variable, then express the length in terms of that variable. Remember that perimeter involves adding all sides of the rectangle.

Show the answer

Answer: width = 5 meters, length = 13 meters

  1. Define variables** Let the width of the garden be \( w \) meters. The length is 3 meters more than twice the width, so: length \( l = 2w + 3 \). --- **
  2. Write the perimeter formula** The perimeter \( P \) of a rectangle is: \( P = 2 \times (\text{length} + \text{width}) \). Given \( P = 36 \), we have: \( 2 \times (l + w) = 36 \). --- **
  3. Substitute the expression for length** Substitute \( l = 2w + 3 \) into the perimeter equation: \( 2 \times ( (2w + 3) + w ) = 36 \). --- **
  4. Simplify inside the parentheses** \( (2w + 3) + w = 3w + 3 \). So: \( 2 \times (3w + 3) = 36 \). --- **
  5. Solve for \( w \)** Divide both sides by 2: \( 3w + 3 = 18 \). Subtract 3 from both sides: \( 3w = 15 \). Divide by 3: \( w = 5 \). --- **
  6. Find the length** \( l = 2w + 3 = 2 \times 5 + 3 = 10 + 3 = 13 \). --- **
  7. Final answer** Width = 5 meters, Length = 13 meters.

Let's go step-by-step. --- **

5 A triangle is drawn on a coordinate plane with vertices at A(1, 2), B(5, 2), and C(3, 6). A line is drawn from point A to the midpoint of side BC. What are the coordinates of this midpoint?

Hint: To find the midpoint between two points, average their x-coordinates and average their y-coordinates separately.

Show the answer

Answer: (4, 4)

  1. Identify the endpoints of side BC: B(5, 2) and C(3, 6).
  2. Calculate the x-coordinate of the midpoint: (5 + 3) ÷ 2 = 8 ÷ 2 = 4.
  3. Calculate the y-coordinate of the midpoint: (2 + 6) ÷ 2 = 8 ÷ 2 = 4.
  4. Combine the coordinates: (4, 4). The midpoint of side BC is (4, 4).

6 A triangle is drawn on a coordinate plane with vertices at A(1, 2), B(5, 6), and C(9, 2). A line is drawn from vertex A to the midpoint of side BC. What are the coordinates of this midpoint?

Hint: To find the midpoint of a line segment, average the x-coordinates and average the y-coordinates of the endpoints. For example, the midpoint between (0,0) and (4,8) would be (2,4).

Show the answer

Answer: (7, 4)

  1. Identify the endpoints of side BC. B is at (5, 6) and C is at (9, 2).
  2. To find the midpoint, average the x-coordinates: (5 + 9) / 2 = 14 / 2 = 7.
  3. Average the y-coordinates: (6 + 2) / 2 = 8 / 2 = 4.
  4. The midpoint of BC is at (7, 4).

The answer is (7, 4).

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