Systems and Relationships Lesson Test

Answer: A

Correct answer: (6, 3). Add the equations: 2x = 12, so x = 6. Then

gives y = 3. The pair (6, 3) checks in both equations.

• A (correct): 6 + 3 = 9 and 6 − 3 = 3, so this pair satisfies both equations.
• B: This pair fits x − y = 3, but 7 + 4 = 11, not 9.
• C: This pair fits x + y = 9, but 5 − 4 = 1, not 3.
• D: The values are in the wrong order; this pair gives 3 − 6 = −3 in the second equation.

Answer: D

Correct answer: Exactly two. Two lines in a plane can cross once (one solution), be parallel (no solution) or be the same line (infinitely many). Straight lines cannot cross in exactly two places, so exactly two solutions is impossible.

• A: Lines with different slopes cross at one point, giving exactly one solution.
• B: Parallel lines never meet, so a system can have no solution.
• C: If both equations describe the same line, every point on it is a solution.
• D (correct): Two straight lines either cross once, never meet or coincide, so two crossing points are impossible.

Answer: D

Correct answer: 15. Replace y with 3x in the second equation:

so 4x = 20 and x = 5. Then

• A: This is the value of x; the question asks for y.
• B: This drops the x term and solves 3x = 20, which ignores part of the second equation.
• C: This multiplies the total 20 by 3 instead of solving for x first.
• D (correct): Substituting gives x + 3x = 20, so x = 5 and y = 3(5) = 15.

Answer: B

Correct answer: (2, 5). Set the right sides equal:

Then 3x = 6, so x = 2, and

The lines cross at (2, 5).

• A: The coordinates are swapped; x must be listed first.
• B (correct): Setting 2x + 1 = −x + 7 gives x = 2, and then y = 5; the point lies on both lines.
• C: This point lies on y = −x + 7, but on the first line x = 3 gives y = 7.
• D: This is only the y-intercept of the second line; the first line passes through (0, 1).

Answer: D

Correct answer: No solution. Setting

leaves

which is false. Equal slopes with different intercepts mean parallel lines, so there is no solution.

• A: Exactly one solution requires different slopes, but both slopes are 4.
• B: The point (0, 1) lies on the first line only; the second line has y = −3 when x = 0.
• C: Infinitely many requires the same line, but the intercepts 1 and −3 differ.
• D (correct): Both lines have slope 4 but different intercepts, so they are parallel and never meet.

Answer: B

Correct answer: (3, 2). The y terms are opposites, so add the equations: 8x = 24 and x = 3. Substitute into the first:

so 2y = 4 and y = 2. The solution is (3, 2).

• A: The values are in the wrong order; this pair gives 10 + 6 = 16 in the first equation.
• B (correct): Adding the equations gives 8x = 24, so x = 3, and 15 + 2y = 19 gives y = 2.
• C: This stops at 2y = 4 and forgets to divide by 2.
• D: This comes from a sign slip in back-substitution; with x = 3, 9 − 2y = 5 gives y = 2.

Answer: A

Correct answer: 3. The y coefficients match, so subtract the second equation from the first: 3x = 12 and x = 4. Then

so 3y = 9 and y = 3.

• A (correct): Subtracting gives 3x = 12, so x = 4; then 8 + 3y = 17 gives y = 3.
• B: This stops at 3y = 9 and forgets to divide by 3.
• C: This is the value of x; the question asks for y.
• D: This uses 2x instead of 5x when back-substituting into the first equation, giving 8 + 3y = 29.

Answer: A

Correct answer: 7. Multiply the second equation by 4:

Subtract the first: 5y = 15, so y = 3. Then

so x = 4, and

• A (correct): Solving gives x = 4 and y = 3, and the question asks for their sum, 7.
• B: This is x − y; the question asks for the sum.
• C: This is only the value of x, not x + y.
• D: This is only the value of y, not x + y.

Answer: C

Correct answer: 12. Let

Substitute

so 5c = 60 and c = 12. Check: 18 adults bring $216 and 12 children bring $84, totaling $300.

• A: This divides $300 by $12 as if every ticket were an adult ticket.
• B: This is the number of adult tickets; the question asks about children.
• C (correct): With a + c = 30 and 12a + 7c = 300, solving gives a = 18 adults and c = 12 children.
• D: This assumes the tickets split evenly, which would bring in only $285.

Answer: D

Correct answer: Infinitely many. Every term of the second equation, 6x, −4y and 16, is twice the matching term of the first. The equations describe the same line, so every point on it is a solution: infinitely many.

• A: No solution requires the constants to break the pattern, but 8 doubled is exactly 16.
• B: Exactly one requires different slopes, but both lines have slope 3/2.
• C: Two straight lines can never meet in exactly two points.
• D (correct): The second equation is exactly twice the first, so both describe the same line.

Answer: C

Correct answer: 23. Let

Adding gives 2x = 82, so the larger number is 41. The smaller is

• A: This is the larger number, (64 + 18)/2.
• B: This subtracts the difference from the sum but forgets to divide by 2.
• C (correct): The smaller number is (64 − 18)/2 = 23; check: 41 + 23 = 64 and 41 − 23 = 18.
• D: This is half the sum, which ignores the difference between the numbers.

Answer: B

Correct answer: (9, 4). Multiply the first equation by 6:

From the second,

which gives 5y = 20 and y = 4. Then x = 9; check:

• A: This multiplies the fractions by 6 but not the constant 5, giving 2x + 3y = 5.
• B (correct): Clearing fractions gives 2x + 3y = 30; with x = y + 5, this yields y = 4 and x = 9.
• C: This multiplies the fractions incorrectly, giving 3x + 2y = 30 instead of 2x + 3y = 30.
• D: The values are reversed; x is 9 and y is 4.

Answer: B

Correct answer: $5. Subtract the second order from the first: 2 drinks = $6, so one drink costs $3. Then

so 2s = 10 and one sandwich costs $5.

• A: This divides $13 by 2 and ignores the drink in the second order.
• B (correct): Subtracting the orders shows 2 drinks cost $6, so a drink is $3; then 2s + 3 = 13 gives s = $5.
• C: This is the cost of two sandwiches; it still needs to be divided by 2.
• D: This is the price of one drink, not a sandwich.

Answer: C

Correct answer: −2. Multiply the first equation by 2: 6x + 2ky = 14. For parallel lines the left sides must match, so

Since 14 does not equal 5, the lines never meet.

• A: This doubles −4 instead of halving it; k must satisfy 2k = −4.
• B: This copies the y coefficient of the second equation without accounting for the doubling of the first.
• C (correct): Doubling the first equation gives 6x + 2ky = 14; with k = −2 the left sides match but 14 is not 5, so the lines are parallel.
• D: With k = 2 the doubled equation has +4y, which does not match −4y, so the lines cross once.

Answer: C

Correct answer: 13. Add the equations: 4x = 16, so x = 4. Then

gives y = 3. Finally,

• A: This is x + y; the question asks for the square of x minus y.
• B: This drops the negative sign on −2 during back-substitution, giving y = 1.
• C (correct): Adding gives x = 4, then y = 3, so 16 − 3 = 13.
• D: This is x − y; x must be squared first.

Answer: B

Correct answer: 15 mph. Downstream speed is

Upstream speed is

Adding gives 2b = 30, so b = 15 mph and the current is 3 mph.

• A: This is the average speed for the round trip, 72 miles in 5 hours, not the still-water speed.
• B (correct): Downstream b + c = 18 and upstream b − c = 12; adding gives 2b = 30, so b = 15 mph.
• C: This is the downstream speed, which includes help from the current.
• D: This is the speed of the current, not the boat.

Answer: C

Correct answer: 20 liters. Let

Multiply the second by 10:

Subtract the first: 3y = 30, so y = 10 and x = 20 liters.

• A: This is the amount of the 40% solution, not the 10% solution.
• B: This is the total amount of salt in the mixture, 20% of 30 liters.
• C (correct): Solving x + y = 30 and 0.1x + 0.4y = 6 gives x = 20 liters of 10% and y = 10 liters of 40%.
• D: Equal amounts would give a 25% mixture, not 20%.

Answer: A

Correct answer:

Compare slopes.

becomes

becomes

The slopes differ, so the lines cross once; each other pair has equal slopes.

• A (correct): Rewritten, the slopes are −2 and 2; different slopes mean the lines cross exactly once.
• B: The variable terms double but 6 doubled is 12, not 5, so the lines are parallel with no solution.
• C: The second equation becomes y = 2x − 0.5, the same slope with a different intercept, so there is no solution.
• D: The second equation is twice the first, so they are the same line with infinitely many solutions.

Answer: B

Correct answer: (3, 5). Isolate x in the first equation:

Substitute into the second:

giving y = 5. Then

and (3, 5) checks in both equations.

• A: This lists the values in the order (y, x); with x = 5 the first equation gives 5 + 6 = 11, not 13.
• B (correct): x = 3 and y = 5 make x + 2y = 13 and 3x − y = 4 both true.
• C: This pair fits x + 2y = 13 but gives 3 − 6 = −3 in the second equation, so it fails the check.
• D: This pair fits 3x − y = 4 but gives 2 + 4 = 6 in the first equation instead of 13.

Answer: D

Correct answer: a = 6 and b = 15. For infinitely many solutions the second equation must be a constant multiple of the first. The y-coefficient goes from −6 to −9, a factor of 1.5. So

making the two equations describe the same line.

• A: Scaling the x-term but not the constant gives parallel lines (same slope, different line), which means no solution.
• B: Copying the first equation's numbers ignores that the y-coefficient changed from −6 to −9, so the slopes would differ and there would be one solution.
• C: This matches the x-coefficient but uses the y-coefficient size as the constant; 10 times 1.5 is 15, so the lines would be parallel.
• D (correct): Multiplying every term of 4x − 6y = 10 by 1.5 gives 6x − 9y = 15, the same line.

Answer: B

Correct answer: 6. Multiply every term of the second equation by 2:

Subtract it from the first equation:

so x = 6. Check:

gives y = 2, and

• A: This is the value of y, not x; the question asks for x.
• B (correct): Doubling x + 2y = 10 and subtracting it from 3x + 4y = 26 leaves x = 26 − 20 = 6.
• C: This comes from doubling only the variable terms and leaving the constant 10, so 26 − 10 = 16; every term must be multiplied.
• D: This is x + y = 6 + 2, the sum of the two values rather than x alone.

Answer: C

Correct answer: $150. Let t be the truck cost and a the ATV cost:

Adding gives

Subtracting the first from the second gives

Adding these, 2t = 300, so t = $150 (and a = $70; check

).

• A: This is the cost of one ATV, not one truck.
• B: This is the combined cost of one truck plus one ATV, found by adding the equations and dividing by 5.
• C (correct): With t = 150 and a = 70, both receipts check: 300 + 210 = 510 and 450 + 140 = 590.
• D: This splits the truck-plus-ATV total of $220 evenly, ignoring that a truck costs $80 more than an ATV.

Answer: A

Correct answer: 1. Subtract the second equation from the first, term by term:

Divide by 2:

(Solving fully gives x = 5.5 and y = 4.5, which agrees.)

• A (correct): Subtracting the equations gives 2x − 2y = 2, and dividing by 2 leaves x − y = 1.
• B: This is x + y, which comes from adding the equations (8x + 8y = 80) rather than subtracting them.
• C: This is the value of 2x − 2y; it still has to be divided by 2 to give x − y.
• D: This is the value of x alone; the question asks for the difference x − y.

Answer: C

Correct answer: The lines are parallel, so the system has no solution. Rewrite both in slope-intercept form. The first becomes

and the second becomes

The slopes are equal but the y-intercepts differ, so the lines are parallel and never meet: the system has no solution.

• A: The variable terms of the first equation are double those of the second, but 8 is not double 6, so the lines are not identical.
• B: The lines would cross once only if their slopes differed; here both slopes are 0.5.
• C (correct): Both lines have slope 0.5 but intercepts −2 and −3, so they never cross.
• D: Perpendicular lines need slopes whose product is −1; these slopes are equal, which makes the lines parallel.

Answer: D

Correct answer: −3. Solve the second equation for y:

For the two equations to describe the same line, their slopes (both 2) and their intercepts must match, so

• A: This drops the sign when dividing 6 by −2; the intercept is −3, and c = 3 would give parallel lines.
• B: This divides only the x-term by −2 and leaves the constant as 6 with a sign change; every term must be divided.
• C: This is the slope of both lines, not the intercept c that must also match.
• D (correct): Rewriting 4x − 2y = 6 gives y = 2x − 3, so c = −3 makes the lines identical.

Answer: A

Correct answer: (3, 12). Substitute y = 4x into the second equation:

so 5x = 15 and x = 3. Then

Check:

• A (correct): With x = 3 and y = 12, both y = 4x and x + y = 15 are true.
• B: This reverses the order; with x = 12, y would have to be 48 for y = 4x.
• C: This pair adds to 15 but 10 is not 4 times 5, so it fits only one equation.
• D: Here y is 4 times x, but 2 + 8 = 10, not 15, so it fits only one equation.

Answer: C

Correct answer: (3, −5). At the crossing point both lines have the same y, so set

and x = 3. Substitute:

The lines meet at (3, −5).

• A: This swaps the coordinates; the x-value must come first in an ordered pair.
• B: This drops the negative sign on y; 3 − 8 equals −5, not 5.
• C (correct): x = 3 gives y = −5 on both lines: −6 + 1 = −5 and 3 − 8 = −5.
• D: This comes from a sign slip when solving −3x = −9; the point lies on the first line but not on y = x − 8.

Answer: B

Correct answer: The point where the two lines intersect. Every point on a line is a solution of that line's equation, so a point that lies on both lines satisfies both equations. That point is where the two lines intersect.

• A: The y-intercept lies on one line only, so it generally does not satisfy the other equation.
• B (correct): Only the crossing point lies on both lines, so only it makes both equations true.
• C: Slopes describe steepness and help classify the system, but they are not an ordered-pair solution.
• D: These x-intercepts each belong to one line and are usually different points.

Answer: C

Correct answer: 44 feet. Let L and S be the lengths:

Adding gives 2L = 88, so L = 44 feet (and S = 28 feet; check

).

• A: This is the shorter piece, (72 − 16)/2, not the longer one.
• B: This subtracts 16 from 72 without splitting the remainder between the two pieces.
• C (correct): Adding L + S = 72 and L − S = 16 gives 2L = 88, so the longer piece is 44 feet.
• D: This cuts the cable in half, ignoring that one piece is 16 feet longer.

Answer: D

Correct answer: It has infinitely many solutions. A true statement with no variables left means the two equations are really the same line. Every point on that line satisfies both, so the system has infinitely many solutions.

• A: No solution is signaled by a false statement such as 0 = 4, not by the true statement 0 = 0.
• B: The zeros come from the variables cancelling, not from the variables equaling zero.
• C: Exactly one solution appears when elimination leaves a single variable with one value, such as x = 4.
• D (correct): A true statement such as 0 = 0 means the equations describe the same line, so every point on it works.

Answer: B

Correct answer: $2. Subtract the second purchase from the first: 2 hinges cost

so one hinge costs $3. Then

gives 2L = 4, so one latch costs $2. Check:

• A: This is the price of one hinge, found in the first step; the question asks for a latch.
• B (correct): With a hinge at $3, the second purchase gives 9 + 2L = 13, so a latch costs $2.
• C: This is the cost of 2 hinges from the subtraction step, not the price of a latch.
• D: This is the price of one hinge plus one latch together, not a latch alone.

Answer: B

Correct answer: 600 gallons per hour. Write

and a = 2b. Substitute:

so 3b = 900 and b = 300. Then

per hour.

• A: This is the rate of Pump B, the slower pump.
• B (correct): Substituting a = 2b into a + b = 900 gives b = 300, so Pump A moves 600 gallons per hour.
• C: This splits the 900 gallons evenly, ignoring that Pump A moves twice as much as Pump B.
• D: This doubles the combined total instead of doubling Pump B's share.

Answer: A

Correct answer: (−4, 7). An ordered pair lists the x-value first and the y-value second, so

and y = 7 is written (−4, 7).

• A (correct): The x-value −4 comes first and the y-value 7 comes second.
• B: This puts y first; reversing the order names a different point.
• C: This moves the negative sign to the wrong coordinate.
• D: This swaps the numbers and keeps the negative in front, giving a different point.

Answer: B

Correct answer: (3, 4). Test each pair in both equations. For (3, 4):

so both are true. A pair that works in only one equation is not a solution of the system.

• A: This adds to 7, but 3(4) − 3 = 9, not 5; it is the solution with the coordinates reversed.
• B (correct): It gives 9 − 4 = 5 and 3 + 4 = 7, so it satisfies both equations.
• C: This fits 3x − y = 5 but adds to only 3, so it fails x + y = 7.
• D: This adds to 7, but 3(5) − 2 = 13, so it fails 3x − y = 5.

Answer: C

Correct answer: 3. Clear the decimals and fractions first. Multiply the first equation by 10:

Multiply the second by 6:

Multiply these by 2 and 5 respectively:

Adding gives 23x = 138, so x = 6, and then

gives y = 3.

• A: This is the value of x; the question asks for y.
• B: This is x + y, the sum of the two values rather than y alone.
• C (correct): With x = 6 and y = 3, both originals check: 2.4 + 1.5 = 3.9 and 3 − 1 = 2.
• D: This comes from a sign slip in 3(6) − 2y = 12; solving correctly gives −2y = −6, so y is positive 3.

Answer: A

Correct answer: (4, −2). The second equation is the first multiplied by 3, so both describe the line

Any point on that line works. For x = 4,

and (4, −2) gives

• A (correct): It satisfies 2(4) + (−2) = 6, and therefore also the tripled equation 6x + 3y = 18.
• B: This reverses the order; 2(−2) + 4 = 0, not 6.
• C: Here 2(2) + 4 = 8, not 6, so this point is not on the shared line.
• D: At x = 0 the line gives y = 6; using y = 3 halves the intercept by mistake, and 0 + 3 is not 6.

Answer: A

Correct answer: Infinitely many. One equation in two unknowns allows endless pairs, such as (2, 10), (5, 7) and (15, −3). Its graph is a whole line of solutions; a second independent equation is needed to narrow the answer to one pair.

• A (correct): Every point on the line x + y = 12, such as (5, 7) or (0, 12), is a solution.
• B: One solution needs a second independent equation; alone, many pairs add to 12.
• C: A linear equation never has exactly two solutions; (1, 11), (6, 6) and (12, 0) already give three.
• D: Many pairs, such as (6, 6), make x + y = 12 true.

Answer: D

Correct answer: 8. Let a be the 15-pound count and b the 25-pound count:

Multiply the first by 15:

Subtract: 10b = 120, so b = 12 and a = 20. The question asks for the difference:

• A: This is the number of 25-pound dumbbells, not the difference the question asks for.
• B: This is the number of 15-pound dumbbells, not how many more there are.
• C: This is 600 divided by 25, which assumes every dumbbell weighs 25 pounds.
• D (correct): There are 20 fifteen-pound and 12 twenty-five-pound dumbbells, so there are 8 more of the lighter ones.

Answer: D

Correct answer: 0 = 5, so the system has no solution. Double the first equation:

Subtract the second equation:

Both variables vanish and a false statement remains, so the lines are parallel and the system has no solution.

• A: The constants do not cancel: 8 − 3 is 5, so the result is false, not 0 = 0.
• B: A false statement means no point works; infinitely many solutions come from a true statement such as 0 = 0.
• C: Both variables cancel here, so no value of x is produced.
• D (correct): Doubling the first equation and subtracting the second leaves the false statement 0 = 5, which signals parallel lines.

Answer: C

Correct answer:

Isolate y. Subtract 6x from both sides:

Divide every term by 3:

so the slope is −2 and the y-intercept is 4.

• A: This loses the negative sign created when 6x moves to the other side.
• B: This divides the x-term by 3 but forgets to divide the constant 12.
• C (correct): Subtracting 6x and dividing every term by 3 gives y = −2x + 4.
• D: This moves 6x across but never divides by the coefficient 3 on y.

Answer: D

Correct answer: (4, 0). Substitute the isolated y:

Distribute:

so 8x = 32 and x = 4. Then

Check:

• A: This comes from writing 3(x − 4) as 3x − 4 without distributing to the 4; then 5(3) + 3(−1) = 12, not 20.
• B: This reverses the coordinates; with x = 0 the first equation gives y = −4.
• C: This comes from a sign slip, using +12 instead of −12; then 5(1) + 3(−3) = −4, not 20.
• D (correct): x = 4 and y = 0 satisfy both y = x − 4 and 5x + 3y = 20.

Answer: A

Correct answer: 5. The x-coefficients match, so subtract the second equation from the first, every term:

That gives 4y = 20, so y = 5. Then

so x = 4.

• A (correct): Subtracting gives 3y − (−y) = 4y and 31 − 11 = 20, so y = 5.
• B: This treats 3y − (−y) as 2y; subtracting a negative adds, giving 4y.
• C: This is the value of x, not y.
• D: This adds the constants (31 + 11 = 42) while subtracting the variable terms; every term must be subtracted.