Tip: Write the answer as (x, y) — the x-value always comes first. Then plug it into both equations; a wrong answer often works in only one.
Question 5 of 36
No calculator
The table shows some values of y for two linear equations, y = 3x + 1 and y = -x + 9.
x
0
1
2
3
4
y = 3x + 1
1
4
7
10
13
y = -x + 9
9
8
7
6
5
What is the solution to the system made of these two equations?
Hint
The solution is the (x, y) pair that is the same for both equations. Look for a column where both rows match.
Step-by-step solution
A solution of the system must have the same y for the same x in both equations.
Scan the columns: at x = 0 the y-values are 1 and 9; at x = 1, 4 and 8; at x = 2, 7 and 7. They match.
So the point (2, 7) is on both lines.
The solution is (2, 7).
Check:3(2) + 1 = 7 ✓ and -2 + 9 = 7 ✓. Algebra agrees: 3x + 1 = -x + 9 gives 4x = 8, so x = 2.
Tip: In a table, the solution is the column where the two output rows are equal. If no column matches, look for where the rows "cross over" (one row goes from smaller to larger than the other) — the solution lies between those columns.
Question 6 of 36
No calculatorFill in
In the system below, the first equation is already solved for x. Use substitution to find the solution, and enter each value.
Tip: After you find one variable, substitute it into whichever original equation is easier. Then check the pair in the other equation — that catches arithmetic mistakes.
Tip: Most two-item word problems give a count fact and a value fact. After solving, make sure you report the item the question asks about.
Question 10 of 36
Drop-down
Select the choice that makes each statement true.
The system \begin{cases} 2x + y = 5 \\ 4x + 2y = 10 \end{cases} has .
The system \begin{cases} y = 3x + 1 \\ y = 3x - 2 \end{cases} has .
Choose an answer from each menu.
Hint
Write each equation in y = mx + b form and compare the slopes and the y-intercepts.
Step-by-step solution
First system.
Solve 2x + y = 5 for y: y = -2x + 5.
Solve 4x + 2y = 10 for y: 2y = -4x + 10, so y = -2x + 5.
The equations describe the same line (same slope -2, same intercept 5). Every point on the line is a solution.
The first system has infinitely many solutions. (Notice the second equation is just the first one multiplied by 2.)
Second system.
Both lines have slope 3, but the intercepts are different (1 and -2). The lines are parallel and never meet.
Algebra agrees: 3x + 1 = 3x - 2 gives 1 = -2, which is false.
The second system has no solution.
Check: in the first system, try (0, 5) and (1, 3) — both work in both equations. In the second system, at any x the two y-values always differ by 3.
Tip: Same slope + same intercept → infinitely many. Same slope + different intercept → none. Different slopes → exactly one.
Question 11 of 36
Fill in
The sum of two numbers is 52. Their difference is 14. What is the larger number?
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Let x be the larger number and y the smaller. Write one equation for the sum and one for the difference, then add them.
Step-by-step solution
Let x = the larger number and y = the smaller number.
Sum: x + y = 52. Difference: x - y = 14.
Add the equations (the y terms cancel): 2x = 66.
Divide by 2: x = 33.
The smaller number is y = 52 - 33 = 19.
The larger number is 33.
Check:33 + 19 = 52 ✓ and 33 - 19 = 14 ✓
Tip: For "sum and difference" problems, adding the two equations always eliminates y. A quick shortcut: larger = \dfrac{\text{sum} + \text{difference}}{2}.
Question 12 of 36
A coffee shop mixes beans that cost $9 per pound with beans that cost $14 per pound. It wants 20 pounds of a blend worth $12 per pound. How many pounds of the $14 beans should it use?
Hint
Write one equation for the total weight and one for the total value. The blend is worth 20 \times 12 dollars.
Step-by-step solution
Let x = pounds of $9 beans and y = pounds of $14 beans.
The shop should use 12 pounds of the $14 beans (and 8 pounds of the $9 beans).
Check:9(8) + 14(12) = 72 + 168 = \$240, and \$240 \div 20 = \$12 per pound ✓
Tip: Estimate first. $12 is closer to $14 than to $9, so there must be more of the $14 beans — more than 10 pounds.
Question 13 of 36
A yoga studio offers two plans:
Plan A: a $30 monthly membership plus $5 per class
Plan B: no membership, $10 per class
Let x be the number of classes in a month and y be the total monthly cost in dollars. Which system can be used to find the number of classes for which the plans cost the same?
Hint
For each plan, which amount is paid once a month, and which is paid for every class?
Step-by-step solution
Plan A: $30 once per month (constant) plus $5 for each class (5x): y = 5x + 30.
Plan B: $10 for each class, nothing else: y = 10x.
The system is \begin{cases} y = 5x + 30 \\ y = 10x \end{cases}
The correct system is y = 5x + 30 and y = 10x.
Check: By solving, 5x + 30 = 10x gives 30 = 5x, so x = 6 classes. Plan A: 5(6) + 30 = \$60; Plan B: 10(6) = \$60 ✓
Tip: In each equation, the per-class price multiplies x and the one-time fee is the constant. Test with x = 1: Plan A should cost $35 and Plan B $10.
Question 14 of 36
No calculator
Adding these two equations makes one variable drop out. Use that idea to decide which point lies on both lines.
\begin{cases} 4x + y = 11 \\ 2x - y = 7 \end{cases}
Hint
The y-terms are +y and -y. What happens if you add the two equations?
Step-by-step solution
The y-terms are opposites (+y and -y), so add the equations:
Tip: Look for a pair of opposite coefficients first — adding makes that variable disappear. Be careful with the second variable: 12 + y = 11 means y is negative.
Question 15 of 36
Which system of equations has no solution?
Hint
Write every equation in y = mx + b form. Lines with no common point are parallel: what do parallel lines have in common, and what is different?
Step-by-step solution
A system has no solution when its lines are parallel: same slope, different y-intercepts.
In every choice the first line is y = 4x - 1 (slope 4, intercept -1).
A:8x - 2y = 6 \Rightarrow -2y = -8x + 6 \Rightarrow y = 4x - 3. Same slope 4, different intercept (-3). Parallel → no solution.
B:8x - 2y = 2 \Rightarrow y = 4x - 1. The same line → infinitely many solutions.
C:y = -4x - 1 has slope -4, which is different from 4 → the lines cross once.
D:y = \tfrac{1}{4}x - 3 has slope \tfrac{1}{4}, different from 4 → one solution.
System A has no solution.
Check: try to solve A: substitute y = 4x - 1 into 8x - 2y = 6: 8x - 8x + 2 = 6, so 2 = 6, which is false for every x ✓
Tip: Same slope + different intercept → no solution. Same slope + same intercept → infinitely many. Different slopes → exactly one, even if the slopes are opposites or reciprocals.
Question 16 of 36
The graph shows the lines 2x + y = 4 and x - y = 5.
Which ordered pair is the solution to the system?
Hint
The solution is where the two lines meet. Count across first (for x), then up or down (for y).
Step-by-step solution
The solution of a system is the point that lies on both lines — the marked crossing point.
Read it: the point is 3 units to the right of the origin and 2 units down, so it is (3, -2).
Confirm with elimination: add the equations, (2x + y) + (x - y) = 4 + 5, so 3x = 9 and x = 3. Then 2(3) + y = 4 gives y = -2.
Tip: Points where a line crosses an axis (like (0, 4) or (5, 0)) are on only one line. The answer must be on both. When a graph point falls below the x-axis, its y-value is negative.
Question 17 of 36
A florist sells roses for $3 each and tulips for $2 each. On Saturday she sold 85 flowers for a total of $215. Let r be the number of roses and t the number of tulips sold. Which system can be used to find r and t?
Hint
Write one equation that counts flowers and one equation that counts dollars.
Step-by-step solution
A word problem with two unknowns needs two equations — one for each fact.
Count of flowers: roses plus tulips is 85: r + t = 85.
Money: each rose brings $3 and each tulip $2, for $215 total: 3r + 2t = 215.
\begin{cases} r + t = 85 \\ 3r + 2t = 215 \end{cases}
The correct system is r + t = 85 and 3r + 2t = 215.
Check: solving gives r = 45, t = 40: 45 + 40 = 85 ✓ and 3(45) + 2(40) = 135 + 80 = 215 ✓
Tip: Label each total with its unit: 85 flowers, $215 dollars. The "count" equation has no prices in it; the "money" equation multiplies each count by its price.
Question 18 of 36
Fill in
The lines y = \tfrac{1}{2}x + 3 and y = -x + 9 intersect at one point. Find the coordinates of that intersection point.
x-coordinate:
y-coordinate:
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Set the two right sides equal. To clear the fraction, you can multiply every term by 2.
Step-by-step solution
Both right sides equal y: \tfrac{1}{2}x + 3 = -x + 9.
Multiply every term by 2 to clear the fraction: x + 6 = -2x + 18.
Tip: Fractions are easier to handle if you clear them first: multiply every term on both sides by the denominator. Without that step, you can also add x to get \tfrac{3}{2}x = 6, then multiply by \tfrac{2}{3}.
Question 19 of 36
Solve this system by first rewriting the top equation so that y is alone. Which ordered pair is the solution?
Tip: When you isolate a variable with a minus sign in front (-y), divide every term by -1. Writing y = 5 - 3x by mistake leads to an answer with very strange numbers — a useful warning sign.
Question 20 of 36
A farm stand sells small baskets of strawberries for $4 and large baskets for $7. One morning it sold 64 baskets and took in $340. How many small baskets were sold?
Hint
Let s = small and \ell = large. Write one equation for the number of baskets and one for the money.
Step-by-step solution
Let s = small baskets, \ell = large baskets.
Baskets: s + \ell = 64
Money: 4s + 7\ell = 340
Multiply the first equation by 4: 4s + 4\ell = 256.
Subtract it from the money equation: (4s + 7\ell) - (4s + 4\ell) = 340 - 256, so 3\ell = 84 and \ell = 28.
Tip: Before you choose an answer, reread which quantity the question asks for. Here the system gives both numbers, and the large-basket count (28) is waiting as a trap.
Question 21 of 36
Select all
Consider the system of equations.
\begin{cases} y = 2x - 1 \\ 3x + y = 9 \end{cases}
Select all the statements that are true.
Select all that apply.
Hint
Find the solution first. Then test each point in each equation separately.
Step-by-step solution
Solve: substitute y = 2x - 1 into 3x + y = 9: 3x + 2x - 1 = 9, so 5x = 10 and x = 2. Then y = 2(2) - 1 = 3. The solution is (2, 3).
A — (2, 3) is a solution of the system: true (shown above).
B — (0, 9): first equation: 2(0) - 1 = -1 \ne 9. It is on the second line only. False.
C — (1, 1) lies on y = 2x - 1:2(1) - 1 = 1 ✓. True.
D — (1, 1) is a solution of the system: second equation: 3(1) + 1 = 4 \ne 9. False.
E — (3, 2) is a solution: first equation: 2(3) - 1 = 5 \ne 2. False.
True statements: A and C.
Check:(2, 3): 2(2) - 1 = 3 ✓ and 3(2) + 3 = 9 ✓
Tip: A point can be on one line without being a solution of the system. "Solution of the system" means it works in every equation.
No variable cancels yet. Multiply each equation by a number so that the y terms become +12y and -12y.
Step-by-step solution
To eliminate y, make the coefficients +12y and -12y. Multiply the first equation by 4 and the second by 3:
\begin{cases} 8x + 12y = 28 \\ 9x - 12y = 57 \end{cases}
Add the equations: 17x = 85, so x = 5.
Back-substitute into 2x + 3y = 7: 10 + 3y = 7, so 3y = -3 and y = -1.
Tip: When you multiply an equation, multiply every term, including the number on the right side. Then check the pair in both original equations.
Question 23 of 36
Fill in
Two companies rent moving trucks for one day.
Company A: $45 plus $0.20 per mile
Company B: $30 plus $0.35 per mile
For what number of miles do the two companies charge the same amount, and what is that cost?
Miles: Cost: $
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Write y = (\text{rate})m + (\text{fee}) for each company. Set the two costs equal to find m, then substitute to find the cost.
Step-by-step solution
Let m = miles and y = cost in dollars.
Company A: y = 0.20m + 45
Company B: y = 0.35m + 30
Set the costs equal: 0.20m + 45 = 0.35m + 30.
Subtract 0.20m: 45 = 0.15m + 30.
Subtract 30: 15 = 0.15m.
Divide: m = \dfrac{15}{0.15} = 100 miles.
Cost: y = 0.20(100) + 45 = 20 + 45 = 65.
At 100 miles, both companies charge $65.
Check: With Company B, 0.35(100) + 30 = 35 + 30 = \$65 ✓
Tip: The solution of a "which plan is cheaper" system is the break-even point. Below 100 miles, Company B (lower fee) is cheaper; above 100 miles, Company A (lower per-mile rate) is cheaper.
Question 24 of 36
Simply adding or subtracting these equations will not cancel a variable — you will need to multiply first. Which point satisfies both equations?
Tip: Multiply every term of an equation, including the number on the right side (10 \times 3 = 30, 18 \times 4 = 72). Forgetting the right side is the most common elimination error.
Question 25 of 36
Drop-down
Select the choices that make the statement true.
The system \begin{cases} 6x - 3y = 9 \\ y = 2x - 3 \end{cases} has because the two lines have .
Choose an answer from each menu.
Hint
Rewrite 6x - 3y = 9 in y = mx + b form and compare it with the second equation.
Step-by-step solution
Solve 6x - 3y = 9 for y: subtract 6x: -3y = -6x + 9. Divide every term by -3: y = 2x - 3.
The second equation is also y = 2x - 3.
Same slope (2) and same y-intercept (-3): the two equations describe the same line. Every point on it solves both equations.
The system has infinitely many solutions because the lines have the same slope and the same y-intercept.
Check: substitute y = 2x - 3 into 6x - 3y = 9: 6x - 6x + 9 = 9, so 9 = 9 — true for every x ✓. For example, (0, -3) and (5, 7) both work.
Tip: When substitution makes the variable disappear, look at what is left: a true statement (9 = 9) means infinitely many solutions; a false one (2 = 6) means no solution.
Question 26 of 36
A lab technician needs 30 liters of a 20% saline solution. She will mix a 10% saline solution with a 40% saline solution. How many liters of the 40% solution should she use?
Hint
Write one equation for the total liters of liquid and one for the liters of pure salt. 20% of 30 liters is how much salt?
Step-by-step solution
Let x = liters of 10% solution and y = liters of 40% solution.
Total liquid:x + y = 30.
Pure salt: the mixture has 0.20 \times 30 = 6 liters of salt, so 0.10x + 0.40y = 6.
From the first equation, x = 30 - y. Substitute: 0.10(30 - y) + 0.40y = 6.
Distribute: 3 - 0.10y + 0.40y = 6, so 0.30y = 3 and y = 10.
Then x = 30 - 10 = 20.
She should use 10 liters of the 40% solution (and 20 liters of the 10% solution).
Tip: 20% is closer to 10% than to 40%, so you need more of the 10% solution — that is a quick way to spot that 10 (not 20) is the amount of the strong solution.
Question 27 of 36
Jordan is 3 times as old as his niece. In 10 years, Jordan will be twice as old as his niece will be then. Let j be Jordan's age now and n be his niece's age now. Which system models this situation?
Hint
In 10 years, both people are 10 years older. Write each future age first, then compare them.
Step-by-step solution
Now: Jordan is 3 times as old as his niece: j = 3n.
In 10 years: Jordan will be j + 10 and his niece will be n + 10.
Jordan will then be twice her age: j + 10 = 2(n + 10).
The correct system is j = 3n and j + 10 = 2(n + 10).
Check: substitute: 3n + 10 = 2n + 20, so n = 10 and j = 30. Today 30 is 3 times 10 ✓. In 10 years, 40 is twice 20 ✓
Tip: Age problems change everyone's age by the same amount. Put the whole future age, n + 10, inside parentheses before multiplying by 2.
Question 28 of 36
The graph shows the total cost of two car-share plans for different numbers of miles driven.
Plan A: $12 plus $0.50 per mile
Plan B: $4 plus $0.90 per mile
At how many miles do the two plans cost the same, and which plan costs less for a 30-mile trip?
Hint
The crossing point is where the costs are equal. Which axis shows miles? To the right of the crossing, which line is lower?
Step-by-step solution
Equal cost: the lines cross at the marked point (20, 22): at 20 miles, both plans cost $22. (The 22 is a cost in dollars, on the vertical axis.)
Confirm with algebra: 12 + 0.50m = 4 + 0.90m \Rightarrow 8 = 0.40m \Rightarrow m = 20, and 12 + 0.50(20) = 22.
30 miles: Plan A costs 12 + 0.50(30) = \$27. Plan B costs 4 + 0.90(30) = \$31. On the graph, the Plan A line is lower to the right of the crossing.
The plans cost the same at 20 miles, and Plan A is cheaper for a 30-mile trip.
Check: at 20 miles: Plan A 12 + 10 = 22, Plan B 4 + 18 = 22 ✓. At 30 miles: 27 < 31 ✓
Tip: A lower starting fee wins for short trips, but a lower per-mile rate wins for long trips. After the break-even point, the plan with the smaller rate (flatter line) is cheaper.
The solution of the system has x = 2. What is the value of a?
a =
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Use the equation without a to find y first.
Step-by-step solution
The second equation has no unknown constant. Substitute x = 2: 3(2) - y = 1, so 6 - y = 1 and y = 5.
The solution is (2, 5). It must also make the first equation true: a(2) + 2(5) = 16.
Simplify: 2a + 10 = 16, so 2a = 6 and a = 3.
a = 3.
Check: the system 3x + 2y = 16, 3x - y = 1: subtract the equations, 3y = 15, so y = 5, and then 3x = 6, x = 2 ✓
Tip: In "find the missing coefficient" problems, the known solution is the key. Find the full point first (both coordinates), then substitute it into the equation with the unknown.
Question 30 of 36
For what value of k does the system below have no solution?
No solution means parallel lines. Write the second equation in y = mx + b form to find its slope.
Step-by-step solution
Solve the second equation for y: 6x - 2y = 5 \Rightarrow -2y = -6x + 5 \Rightarrow y = 3x - 2.5.
Its slope is 3 and its y-intercept is -2.5.
No solution means the lines are parallel: equal slopes, different intercepts. The first line has intercept 4 (different from -2.5), so we only need k = 3.
k = 3.
Check: with k = 3, substitute y = 3x + 4 into 6x - 2y = 5: 6x - 6x - 8 = 5, so -8 = 5 — false for every x, so there is no solution ✓
Tip: The slope is the coefficient of xonly after the equation is solved for y. In 6x - 2y = 5 the slope is \tfrac{6}{2} = 3, not 6.
Question 31 of 36
Hot spot
Two lines are graphed on the coordinate plane below: a blue line, y = \tfrac{1}{2}x + 3, and an orange line, y = -2x - 2.
Select the point on the graph whose coordinates satisfy both equations.
Click on the figure to mark your answer (or focus it and use the arrow keys).
Hint
Find the one point that lies on both lines. You can confirm your choice by plugging it into both equations.
Step-by-step solution
The solution of a system is the point where the two lines cross.
From the graph, the lines cross 2 units left of the origin and 2 units up: (-2, 2).
Confirm with algebra: \tfrac{1}{2}x + 3 = -2x - 2. Multiply every term by 2: x + 6 = -4x - 4. Add 4x: 5x + 6 = -4. Subtract 6: 5x = -10, so x = -2.
Tip: The y-intercepts (0, 3) and (0, -2) are each on only one line, so they cannot be the solution. For a point left of the y-axis, the x-coordinate is negative.
Question 32 of 36
Consider the point (-1.5,\ 4) and the system of equations below.
Substitute x = -1.5 and y = 4 into each equation separately. Be careful with the negative signs.
Step-by-step solution
Test the point in each equation, with x = -1.5 and y = 4.
First equation:4(-1.5) + 4 = -6 + 4 = -2. The right side is -2 ✓
Second equation:2(-1.5) - 3(4) = -3 - 12 = -15. The right side is -15 ✓
The point makes both equations true, so it is the solution of the system.
(-1.5, 4) is the solution of the system.
Check: solve directly. From the first equation, y = -4x - 2. Substitute: 2x - 3(-4x - 2) = -15 \Rightarrow 2x + 12x + 6 = -15 \Rightarrow 14x = -21 \Rightarrow x = -1.5, and y = 6 - 2 = 4 ✓
Tip: Testing a point is often faster than solving. Write each substitution with parentheses — 4(-1.5), -3(4) — so the signs stay correct.
Question 33 of 36
A boat travels 36 miles downstream (with the current) in 2 hours. The return trip of 36 miles upstream (against the current) takes 3 hours. The boat's speed in still water and the speed of the current stay the same. What is the speed of the current, in miles per hour?
Hint
Downstream, the current adds to the boat's speed; upstream, it subtracts. Use d = rt to find each speed, then set up two equations.
Step-by-step solution
Let b = boat speed in still water and c = current speed (mph).
Downstream speed = \dfrac{36}{2} = 18 mph, and it equals b + c: b + c = 18.
Upstream speed = \dfrac{36}{3} = 12 mph, and it equals b - c: b - c = 12.
Tip: With/against problems (current, wind) always give a sum and a difference. Adding the equations finds the boat (or plane) speed; subtracting finds the current (or wind) speed. Read which one the question asks for.
Tip: Decimals in a system are not harder — clear them first by multiplying by 10 (or 100). Then use the same elimination steps as always.
Question 36 of 36
Fill in
A club is ordering printed T-shirts. Two shops give these prices:
Shop A: $75 setup fee plus $6.50 per shirt
Shop B: no setup fee, $9.00 per shirt
What is the least number of shirts the club must order for Shop A to be cheaper than Shop B?
shirts
Type your answer in the box. Fractions like 3/4 are fine.
Hint
First find the number of shirts where the two shops cost exactly the same. Then think about which shop is cheaper for one more shirt.
Step-by-step solution
Write the costs for n shirts: Shop A: C = 75 + 6.50n. Shop B: C = 9n.
Break-even: set them equal: 75 + 6.50n = 9n. Subtract 6.50n: 75 = 2.50n, so n = 30.
At 30 shirts both cost the same: 75 + 6.50(30) = 75 + 195 = \$270 and 9(30) = \$270.
Shop A has the smaller per-shirt price, so it becomes cheaper after the break-even point. The least whole number greater than 30 is 31.
The club must order at least 31 shirts.
Check: 31 shirts: Shop A 75 + 6.50(31) = 75 + 201.50 = \$276.50; Shop B 9(31) = \$279. Shop A is cheaper ✓. At 30 shirts they tie, so 30 is not "cheaper" ✓
Tip: "Cheaper than" is strict: at the break-even number the costs are equal, so you need the next whole number. Always check one value on each side of the break-even point.
