Algebra 1 Practice Test: Systems of Equations and Inequalities
- Systems of Equations and Inequalities
- 15 questions
- 20 minutes
These 15 questions practice solving systems by substitution, elimination and graphing, deciding how many solutions a system has, writing systems from word problems, systems of inequalities and a few linear-quadratic systems. Check a solution by substituting it into every equation of the system.
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Question 1
3x + 2y = 16
5x − 2y = 0What is the solution (x, y) of the system of equations shown?
Correct answer: D
Adding the equations eliminates y: 8x = 16, so x = 2. Then 3(2) + 2y = 16 gives y = 5. The point (4, 2) satisfies only the first equation.
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Question 2
y = 2x − 3
4x − 3y = 1If (x, y) is the solution of the system, what is x + y?
Correct answer: C
Substitute: 4x − 3(2x − 3) = 1, so −2x + 9 = 1 and x = 4. Then y = 2(4) − 3 = 5, and x + y = 9. The value 1 is y − x.
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Question 3
How many solutions does the system 2x − 4y = 8 and x − 2y = 4 have?
Correct answer: D
Multiplying x − 2y = 4 by 2 gives 2x − 4y = 8, the same equation. Both equations describe the same line, so every point on it is a solution.
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Question 4
For what value of k does the system y = kx + 3 and y = 4x − 1 have no solution?
Correct answer: A
Two lines have no common point when they are parallel and distinct: equal slopes, different intercepts. With k = 4 the slopes match and the intercepts 3 and −1 differ.
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Question 5
A school play sold 150 tickets for a total of $1,390. Adult tickets cost $12 and student tickets cost $7. How many student tickets were sold?
Correct answer: C
Let a + s = 150 and 12a + 7s = 1390. Then 12(150 − s) + 7s = 1390, so 1800 − 5s = 1390 and s = 82. The number 68 is the adult tickets.
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Question 6
A chemist mixes a 20% acid solution with a 50% acid solution to make 30 liters of a 30% acid solution. How many liters of the 50% solution does she use?
Correct answer: A
Let a be liters of the 20% solution: 0.2a + 0.5(30 − a) = 0.3(30), so 15 − 0.3a = 9 and a = 20. The 50% solution is 30 − 20 = 10 liters; 20 is the amount of the 20% solution.
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Question 7
The graphs of two linear equations are shown.
What is the solution of the system?
Correct answer: A
The solution is the point where the lines cross, (2, 1). Check with the equations of the lines, y = x − 1 and y = −2x + 5: both give y = 1 at x = 2. The point (1, 2) swaps the coordinates.
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Question 8
Which point is a solution of the system y > 2x − 1 and y ≤ −x + 4?
Correct answer: D
For (0, 2): 2 > −1 and 2 ≤ 4, so both are true. For (2, 3): 3 > 3 is false because the first inequality is strict. (3, 2) fails the first inequality and (−1, 6) fails the second.
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Question 9
A club sells cookies for $2 each and brownies for $3 each. It needs to raise at least $120 and can bake at most 50 items in total. Let c be the number of cookies and b the number of brownies. Which system models the situation?
Correct answer: B
"At least $120" means the money raised, 2c + 3b, is ≥ 120. "At most 50 items" means c + b ≤ 50. Swapping the prices or the two limits gives a different situation.
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Question 10
2x + 3y = 4
3x − 4y = 23If (x, y) is the solution of the system, what is x − y?
Correct answer: B
Multiply the first equation by 4 and the second by 3: 8x + 12y = 16 and 9x − 12y = 69. Adding gives 17x = 85, so x = 5 and y = −2. Then x − y = 5 − (−2) = 7; 3 is x + y.
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Question 11
Which ordered pair is a solution of the system y = x2 and y = x + 6?
Correct answer: A
Set x2 = x + 6: x2 − x − 6 = 0, so (x − 3)(x + 2) = 0. The solutions are (3, 9) and (−2, 4). (−3, 9) is on the parabola but not on the line, since −3 + 6 = 3.
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Question 12
How many solutions does the system y = x2 and y = 2x − 1 have?
Correct answer: B
Set x2 = 2x − 1: x2 − 2x + 1 = 0, which is (x − 1)2 = 0. There is only one solution, (1, 1): the line touches the parabola at a single point.
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Question 13
The sum of two numbers is 41 and their difference is 13. What is the larger number?
Correct answer: B
From x + y = 41 and x − y = 13, adding gives 2x = 54, so x = 27 and y = 14. The number 54 is 2x, before dividing by 2.
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Question 14
A boat travels 36 km downstream in 2 hours and returns the same 36 km upstream in 3 hours. What is the speed of the boat in still water, in km per hour?
Correct answer: C
Downstream b + c = 18, upstream b − c = 12. Adding gives 2b = 30, so b = 15 km/h and the current is 3 km/h. The values 18 and 12 are the downstream and upstream speeds.
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Question 15
4x + 3y = 10
5x − 2y = 1Which step eliminates y from this system?
Correct answer: C
Multiplying gives 8x + 6y = 20 and 15x − 6y = 3. The y-terms are opposites, so adding removes them. Subtracting the same two equations would give 12y, not 0.
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