Quiz: Systems of Equations and Inequalities
Test your understanding of solving systems of equations and inequalities using multiple methods with these questions.
1. What is a solution to a system of two linear equations?
- The y-intercept of both lines
- The slope of both lines
- An ordered pair that satisfies both equations
- Any point on either line
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The correct answer is C. A solution to a system of equations is an ordered pair (x, y) that makes both equations true simultaneously. Graphically, it's the point where the two lines intersect. Option A would only be a solution if both lines have the same y-intercept and are the same line. Option B describes parallel lines. Option D satisfies only one equation.
Concept Tested: Solution of a System
2. A system of equations has no solution. What does this mean graphically?
- The lines intersect at one point
- The lines are the same line
- The lines are parallel
- The lines are perpendicular
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The correct answer is C. When a system has no solution, it is called an inconsistent system. Graphically, this means the two lines are parallel—they have the same slope but different y-intercepts, so they never intersect. Option A describes one solution. Option B describes infinitely many solutions. Option D doesn't relate to number of solutions.
Concept Tested: Inconsistent System
See: Types of Systems
3. Solve by substitution: y = 2x + 1 and 3x + y = 11. What is x?
- x = 1
- x = 2
- x = 3
- x = 4
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The correct answer is B. Substitute y = 2x + 1 into the second equation: 3x + (2x + 1) = 11. Simplify: 5x + 1 = 11, so 5x = 10, giving x = 2. Then y = 2(2) + 1 = 5. The solution is (2, 5). Option A would give y = 3, which doesn't satisfy 3x + y = 11. Options C and D also don't satisfy both equations.
Concept Tested: Substitution Method
4. Which method is best for solving this system: x + y = 10 and x - y = 4?
- Graphing
- Substitution
- Elimination
- All methods are equally efficient
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The correct answer is C. The elimination method is most efficient here because the y-coefficients are already opposites (+1 and -1). Adding the equations eliminates y immediately: 2x = 14, so x = 7. Then y = 3. While all methods work, elimination is fastest for this system. Substitution would require solving for a variable first. Graphing is less precise.
Concept Tested: Elimination Method / Choosing the Best Method
5. What type of system has infinitely many solutions?
- Inconsistent system
- Independent system
- Dependent system
- Conditional system
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The correct answer is C. A dependent system has infinitely many solutions because the two equations represent the same line. Every point on the line is a solution. Option A has no solution. Option B has exactly one solution. Option D is not a standard classification for systems.
Concept Tested: Dependent System
6. Solve using elimination: 2x + 3y = 12 and 4x - 3y = 6. What is the solution?
- (3, 2)
- (2, 3)
- (6, 0)
- (0, 4)
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The correct answer is A. Add the equations to eliminate y: (2x + 3y) + (4x - 3y) = 12 + 6, giving 6x = 18, so x = 3. Substitute into the first equation: 2(3) + 3y = 12, so 6 + 3y = 12, giving 3y = 6 and y = 2. The solution is (3, 2). You can verify: 4(3) - 3(2) = 12 - 6 = 6 ✓
Concept Tested: Elimination Method / Linear Combination
7. When graphing a system of inequalities, what represents the solution?
- The boundary lines only
- The region where shadings overlap
- Any point on either boundary line
- The area outside both inequalities
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The correct answer is B. The solution region for a system of inequalities is where the shaded regions of all inequalities overlap. Any point in this overlapping region satisfies all inequalities in the system. Option A represents only the boundaries, not the solution region. Option C may or may not be solutions depending on whether inequalities use ≤/≥ or symbols. Option D describes non-solutions.
Concept Tested: System of Inequalities / Solution Region
8. For the inequality y ≤ 2x + 1, should the boundary line be solid or dashed?
- Solid, because of ≤
- Dashed, because it's an inequality
- Solid, because the slope is positive
- Dashed, because of ≤
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The correct answer is A. The ≤ symbol means "less than or equal to," so points on the line y = 2x + 1 are included in the solution. We use a solid line to show inclusion. A dashed line is used for < or > (strict inequalities) where the boundary is not included. The slope doesn't determine line style.
Concept Tested: Boundary Line / Graphing Systems
9. A movie theater sells adult tickets for $12 and student tickets for $8. They sold 150 tickets for $1,520. Which system models this?
- a + s = 150; 12a + 8s = 1520
- a + s = 1520; 12a + 8s = 150
- 12a + 8s = 150; a + s = 1520
- a - s = 150; 12a - 8s = 1520
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The correct answer is A. Let a = adult tickets and s = student tickets. The total number of tickets is a + s = 150. The total revenue is 12a + 8s = 1520 (adult tickets at $12 each plus student tickets at $8 each). Option B reverses the values. Option C also reverses them. Option D uses subtraction instead of addition.
Concept Tested: Applications of Systems
10. What happens when solving a dependent system algebraically?
- You get a false statement like 0 = 5
- You get a true statement like 0 = 0
- You get exactly one solution
- You get two different solutions
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The correct answer is B. When solving a dependent system (same line, infinitely many solutions), all variables cancel out and you're left with a true statement like 0 = 0 or 5 = 5. This indicates every point on the line is a solution. Option A describes an inconsistent system (no solution). Options C and D describe independent systems.
Concept Tested: Dependent System / Special Cases