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Linear Equation Solutions
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Linear Equation Solutions
Solve each equation. Identify whether it has one solution, no solution, or infinitely many solutions. Show your reasoning where space is provided.
Identify the Solution Type
1.Solve 7x − 4 = 3x + 16. How many solutions does the equation have?
2.Solve 6(x + 2) = 6x + 9. How many solutions does the equation have?
3.Solve 5(2x − 3) = 10x − 15. How many solutions does the equation have?
4.An equation simplifies to 11 = 4. What does this tell you about its solution set?
- It has exactly one solution.
- It has no solution.
- It has infinitely many solutions.
5.An equation simplifies to 0 = 0. What does this tell you about its solution set?
- It has exactly one solution.
- It has no solution.
- It has infinitely many solutions.
Equations in Context
6.At a skate park, Ramp A charges a $4 entry fee plus $2 per ride. Ramp B charges a $10 entry fee plus $2 per ride. Let r be the number of rides and C the total cost in dollars. Write a cost equation for each ramp.
7.Using C = 2r + 4 for Ramp A and C = 2r + 10 for Ramp B, set the costs equal and determine whether any ride count makes the costs equal.
8.Explain what the result in the previous question means about the two skate park costs, for any number of rides.
9.Two skate park memberships each cost $18 to join and $3 per session. Let s be the number of sessions and C the total cost. Write both cost equations and state the solution type when you set them equal.
Graphs and Meaning
10.Match each description of a pair of lines to its equation solution type: one solution, no solution, or infinitely many solutions.
- Graph A: Two distinct parallel lines
- Graph B: Two lines that intersect at exactly one point
- Graph C: Two lines that lie exactly on top of each other
- Infinitely many solutions
- No solution
- One solution
11.In a real-world comparison, what does it mean if the equation for finding when two quantities are equal has no solution?
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