Linear Functions and Systems Review — High School Algebra 2 practice questions

A system of two linear equations whose graphs are parallel lines has:

1. no solution
2. exactly one solution
3. infinitely many solutions
4. two solutions

Answer: A — no solution

A) Correct — Parallel lines never meet. B) Requires lines that cross. C) Requires the same line. D) Two distinct lines meet at most once.

The equations 2x + 4y = 8 and x + 2y = 4 form a system with:

1. no solution
2. exactly one solution
3. the solution (0, 0)
4. infinitely many solutions

Answer: D — infinitely many solutions

D) Correct — The first equation is twice the second, so they are the same line. A) Would need parallel, different lines. B) The lines coincide. C) (0, 0) doesn't satisfy either equation.

Solve the system y = 2x and 3x + y = 15.

1. (6, 3)
2. (5, 10)
3. (3, 9)
4. (3, 6)

Answer: D — (3, 6)

D) Correct — Substitute: 3x + 2x = 15, so x = 3 and y = 6. A) Swaps x and y. B) Divides 15 by 3 instead of 5. C) Finds y from 3x instead of 2x.

Solve −2x + 1 > 7.

1. x > −3
2. x < −3
3. x < 3
4. x > 3

Answer: B — x < −3

B) Correct — Dividing by −2 flips the inequality. A) Forgets to flip the inequality. C) Drops the negative sign. D) Drops the sign and doesn't flip.

What is the vertex of y = |x − 2| + 3?

1. (−2, 3)
2. (2, −3)
3. (3, 2)
4. (2, 3)

Answer: D — (2, 3)

D) Correct — y = |x − h| + k has vertex (h, k). A) Gets the sign of the horizontal shift wrong. B) Gets the sign of the vertical shift wrong. C) Swaps the coordinates.

What is the slope of 2x + 3y = 12?

1. 2/3
2. −2/3
3. −2
4. 4

Answer: B — −2/3

B) Correct — Solve for y: y = −(2/3)x + 4. A) Forgets the negative sign. C) Doesn't divide by 3. D) Is the y-intercept.

What is the slope of a line perpendicular to y = (2/3)x + 1?

1. 3/2
2. −2/3
3. 2/3
4. −3/2

Answer: D — −3/2

D) Correct — Perpendicular slopes are negative reciprocals. A) Takes the reciprocal but forgets the negative. B) Changes the sign but forgets the reciprocal. C) Is the slope of a parallel line.

Solve |x − 3| = 5.

1. x = 8 or x = −2
2. x = 8 only
3. x = 8 or x = −8
4. x = −8 or x = 2

Answer: A — x = 8 or x = −2

A) Correct — x − 3 = 5 or x − 3 = −5. B) Forgets the negative case. C) Negates the first answer instead of solving x − 3 = −5. D) Reverses the signs.

What is the inverse of f(x) = 2x + 3?

1. f⁻¹(x) = (x + 3)/2
2. f⁻¹(x) = 2x − 3
3. f⁻¹(x) = 1/(2x + 3)
4. f⁻¹(x) = (x − 3)/2

Answer: D — f⁻¹(x) = (x − 3)/2

D) Correct — Swap x and y: x = 2y + 3, then solve for y. A) Adds 3 instead of subtracting. B) Only changes the sign of the constant. C) Takes the reciprocal, not the inverse function.

What is the slope of the line through (1, 2) and (4, 11)?

1. 1/3
2. 9
3. 13/5
4. 3

Answer: D — 3

D) Correct — m = (11 − 2)/(4 − 1) = 9/3. A) Divides the run by the rise. B) Uses the rise without dividing by the run. C) Adds the coordinates instead of subtracting.

What's the 'range' of a function?

The set of all outputs (y-values).

Solve by elimination: x + y = 10 and x − y = 2.

(6, 4).

What is f(g(x)) called?

Composition of functions. Plug g(x) into f.

Solve |x − 1| = 5.

x = 6 or x = −4.