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Quizzes > High School Quizzes > Mathematics

Unit 2: Linear Functions Practice Quiz

Study guide & answer key for success

Difficulty: Moderate
Grade: Grade 8
Study OutcomesCheat Sheet
Paper art promoting Linear Systems Showdown quiz for high school Algebra students.

Solve the system: x + y = 10 and x - y = 2.
(6, 4)
(8, 2)
(5, 5)
(7, 3)
By adding the two equations, the y terms cancel out, giving 2x = 12 and thus x = 6. Substituting back into one of the original equations yields y = 4.
What is the slope of the line given by the equation y = 2x + 3?
2x
The y-intercept is 3
3
2
The line is in slope-intercept form, y = mx + b, where m is the slope. Here, m = 2, which is the coefficient of x.
Which of the following systems of equations represents parallel lines (i.e., has no solution)?
x + y = 5 and x - y = 1
x + y = 5 and 2x - y = 7
x + y = 5 and 2x + 2y = 12
x + y = 5 and 2x + 2y = 10
In the correct option, if you attempt to scale the first equation, you would expect 2x + 2y = 10; since 10 does not equal 12, the lines are parallel and do not intersect.
Convert the equation 2y = 4x + 6 into slope-intercept form.
y = 2x - 3
y = 4x + 6
y = 2x + 3
y = x + 3
Dividing every term in 2y = 4x + 6 by 2 yields y = 2x + 3, which is the slope-intercept form of the equation.
In the slope-intercept form of a line, y = mx + b, what does the term b represent?
(b, m)
(m, 0)
(0, b)
(1, m)
In the equation y = mx + b, the term b is the y-intercept, which is the point where the line crosses the y-axis. This occurs when x is zero, so the point is (0, b).
Solve the system using substitution: y = 3x - 2 and 2x + y = 8.
(2, 4)
(3, 2)
(2, 3)
(3, 4)
Substitute y = 3x - 2 into 2x + y = 8 to obtain 2x + 3x - 2 = 8, which simplifies to 5x = 10. Thus, x = 2 and substituting back gives y = 4.
Solve the system using elimination: 3x + 2y = 12 and 5x - 2y = 8.
(2, 9/4)
(5/2, 3/2)
(5/2, 9/4)
(3, 2)
Adding the equations eliminates y (2y and -2y cancel), resulting in 8x = 20, so x = 5/2. Substituting back into one equation leads to y = 9/4.
Solve using elimination: 2x - y = 3 and x + y = 5.
(7/3, 8/3)
(8/3, 7/3)
(2, 3)
(3, 2)
Adding 2x - y = 3 and x + y = 5 cancels y, giving 3x = 8 so that x = 8/3. Substituting back, y = 5 - 8/3 = 7/3.
A line passes through the point (2,3) and has a slope of 4. What is its y-intercept?
2
3
-5
4
Using the point-slope form, substitute the point (2,3) and slope 4 into y - 3 = 4(x - 2) to obtain y = 4x - 5. Thus, the y-intercept is -5.
Which of the following systems of equations has infinitely many solutions?
y = 2x + 3 and y = -2x + 3
y = 2x + 3 and y = 2x - 2
y = 2x + 3 and 2y = 4x + 6
y = 2x + 3 and 2y = 4x + 5
In the correct system, the second equation is simply a multiple of the first, meaning they represent the same line. This leads to infinitely many solutions.
Solve using substitution: x + y = 7 and y = 2x + 1.
(4, 3)
(2, 5)
(3, 4)
(5, 2)
Substitute y = 2x + 1 into x + y = 7 to get x + 2x + 1 = 7, which simplifies to 3x = 6. Thus, x = 2 and then y = 5.
Solve the system using elimination: 4x - y = 9 and 2x + y = 5.
(1/3, 7/3)
(2, 1)
(3, 2)
(7/3, 1/3)
Adding the two equations cancels out y, resulting in 6x = 14, so x = 7/3. Substituting back into one of the equations gives y = 1/3.
If the system 2x + 3y = k and 4x + 6y = 12 has infinitely many solutions, what must k be?
0
6
3
12
For the two equations to be equivalent, the second must be a multiple of the first. Multiplying 2x + 3y = k by 2 gives 4x + 6y = 2k; setting 2k equal to 12 shows that k = 6.
Find the intersection point of the lines y = -x + 4 and y = 2x - 1.
(5/3, 7/3)
(3, 1)
(1, 3)
(7/3, 5/3)
Setting -x + 4 equal to 2x - 1 gives -x - 2x = -1 - 4, so x = 5/3. Substituting back into one of the equations gives y = 7/3.
Which method is most efficient to solve a system when one equation already has a variable isolated?
Substitution
Graphing
Using matrices
Elimination
When one variable is already isolated, substitution is the simplest method since you can directly replace that variable in the other equation.
Solve the system using elimination: 5x + 2y = 1 and 3x - 4y = 11.
(-2, -1)
(-1, 2)
(2, 1)
(1, -2)
Multiplying the first equation by 2 gives 10x + 4y = 2, which when added to the second equation eliminates y. This leads to 13x = 13, so x = 1 and subsequently y = -2.
Solve the system: (1/2)x + (1/3)y = 5 and (1/4)x - (1/6)y = 1.
(7/2, 9)
(7, 2)
(7, 9/2)
(14, 9)
Multiplying the first equation by 6 yields 3x + 2y = 30, and multiplying the second by 12 yields 3x - 2y = 12. Adding these gives 6x = 42, so x = 7. Substituting back gives y = 9/2.
Determine the values of m for which the system y = mx + 4 and y = 2x - 1 has exactly one solution.
Only m = 2
Any real number
Any real number except 2
Only m ≠ 0
For the two lines to intersect at exactly one point, their slopes must be different. Hence, m can be any real number as long as it is not equal to 2.
For which value of c does the system x + 2y = 3 and 2x + 4y = c have infinitely many solutions?
6
9
3
0
Multiplying the first equation by 2 yields 2x + 4y = 6. For the two equations to be equivalent, c must equal 6.
Solve by substitution: y = (1/2)x - 3 and 3x - 2y = 12.
(3, -1.5)
(3, 1.5)
(-3, -1.5)
(-3, 1.5)
Substitute y = (1/2)x - 3 into 3x - 2y = 12 to obtain 3x - [x - 6] = 12, which simplifies to 2x + 6 = 12, hence x = 3. Substituting back gives y = -1.5.
0
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Study Outcomes

  1. Analyze linear equations and their graphical representations to understand systems.
  2. Apply substitution and elimination methods to solve systems of linear equations.
  3. Evaluate the consistency and uniqueness of solutions within linear systems.
  4. Interpret the results of solving linear systems in relation to real-world scenarios.

Unit 2 Test Study Guide: Linear Functions & Systems Cheat Sheet

  1. Recognize different solution types - A system of linear equations can have exactly one solution, no solution, or infinitely many solutions; understanding this helps you know what to expect before you even start solving. For instance, two lines that cross once give one solution, parallel lines give none, and overlapping lines give endless solutions. Maths Is Fun: Systems of Linear Equations
  2. Master the substitution method - Solve one equation for a variable and then plug that expression into the other equation; this is super handy when one variable is easy to isolate. It's like swapping out ingredients in a recipe to make everything fit perfectly. Maths Is Fun: Substitution
  3. Use the elimination method - Add or subtract equations to cancel out one variable, leaving you with a single-variable equation that's a breeze to solve. Think of it as a mathematical tug‑of‑war where one side gives in and goes away! Maths Is Fun: Elimination
  4. Solve systems graphically - Plot each equation on the same axes; the intersection point is your solution. This visual approach helps you "see" why some systems have no solution or infinitely many - perfect for learners who love pictures! Maths Is Fun: Graphical Method
  5. Interpret line relationships - Parallel lines mean no solution, intersecting lines mean one, and identical lines mean infinite solutions. Imagining these scenarios on a coordinate grid cements your conceptual understanding. Maths Is Fun: Types of Solutions
  6. Explore Gaussian elimination - Use systematic row operations to reduce your system into a simpler form (row echelon form) and then solve. This powerful algorithm is the gateway to advanced linear algebra and helps automate solutions for large systems. Wikipedia: Gaussian elimination
  7. Apply to real‑world problems - Systems pop up everywhere, from figuring out when two runners meet to balancing budgets. Tackling relatable word problems boosts your engagement and shows how math powers everyday decisions. Maths Is Fun: Real‑World Applications
  8. Try interactive tools - Platforms like GeoGebra let you manipulate lines in real time and see how solutions change. Immediate visual feedback takes your practice from passive worksheets to active exploration. GeoGebra: Systems Solver
  9. Check determinants for clues - In a two‑equation system, the determinant of the coefficient matrix tells you if there's a unique solution (non‑zero) or if you're dealing with no or infinite solutions (zero). It's a quick algebraic shortcut! Wikipedia: Determinant
  10. Practice makes perfect - The more varied systems you solve, the more confident and flexible you become at spotting the best method. Keep tackling new problems, join study groups, or challenge yourself with timed quizzes to level up fast. Khan Academy: Practice Problems
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