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Quizzes > Mathematics

Linear Equation Word Problems and Linear Programming Quiz

Quick, free linear equations practice quiz with instant results and tips.

Editorial: Review CompletedCreated By: Burza AlexUpdated Aug 23, 2025
Difficulty: Moderate
Grade: Grade 8
Study OutcomesCheat Sheet
Paper art representing Equation and Optimization Challenge quiz for high school and college students.

This quiz helps you practice linear equation word problems and simple linear programming, from translating stories to writing equations and choosing the best solution. For extra review, work through the linear equations quiz, revisit the slope-intercept form quiz, or sharpen with the linear systems practice quiz.

Solve for x: 2x + 3 = 11.
2
4
5
8
Subtracting 3 from both sides gives 2x = 8 and then dividing by 2 yields x = 4. This direct approach confirms that 4 is the correct solution.
Solve for x: x - 5 = 10.
15
20
5
10
Adding 5 to both sides of the equation gives x = 15. The operation directly isolates the variable.
Find the value of y if 3y = 12.
6
4
3
12
Dividing both sides of the equation by 3 results in y = 4. This is a straightforward division problem.
Solve for x: 4x = 20.
4
6
10
5
Dividing both sides by 4 gives x = 5. The solution is obtained by directly isolating x.
Solve for x: x/3 = 2.
5
2
3
6
Multiplying both sides by 3 results in x = 6. This applies the inverse operation of division.
Solve for x: 2x - 4 = 10.
7
8
6
14
Adding 4 to both sides gives 2x = 14, and then dividing by 2 results in x = 7. This is a simple linear equation.
Solve for x: 3(x + 2) = 18.
3
5
6
4
Dividing both sides by 3 simplifies the equation to x + 2 = 6, and subtracting 2 yields x = 4. This demonstrates the distributive property in reverse.
Solve the equation: 5x + 2 = 3x + 10.
4
5
6
3
Subtracting 3x from both sides gives 2x + 2 = 10, and then subtracting 2 leads to 2x = 8, so x = 4. This reinforces combining like terms.
If 6x = 48, what is the value of x?
6
8
48
7
Dividing 48 by 6 directly yields x = 8. This problem tests basic division skills.
Solve for y: 7y - 14 = 0.
7
2
0
14
Adding 14 to both sides results in 7y = 14, and dividing by 7 gives y = 2. This equation is solved by isolating the variable.
Solve for x: 2(x - 3) = 10.
10
13
5
8
Dividing both sides by 2 yields x - 3 = 5, and adding 3 results in x = 8. This step-by-step isolation of x is key in solving the equation.
Solve the equation: 4(x + 1) - 2(x - 3) = 10.
2
-1
0
1
Expanding gives 4x + 4 - 2x + 6 which simplifies to 2x + 10 = 10. Subtracting 10 from both sides leads to 2x = 0, so x = 0.
Solve for x: x/2 + 3 = 7.
8
10
4
7
Subtracting 3 from both sides results in x/2 = 4; then multiplying by 2 yields x = 8. This demonstrates the inverse operation of addition.
Solve for x: 3x + 4 = 2x + 10.
4
6
7
10
Subtracting 2x from both sides gives x + 4 = 10, and subtracting 4 yields x = 6. This is a basic one-variable linear equation.
If 8x - 4 = 4x + 12, what is the value of x?
4
6
5
3
Subtracting 4x from both sides gives 4x - 4 = 12; then adding 4 yields 4x = 16 and dividing by 4 results in x = 4. The steps clearly isolate x.
A rectangle's length is 3 times its width. If the perimeter is 64 units, what is the width?
10
12
16
8
The perimeter of a rectangle is given by 2(length + width). Setting length as 3w leads to 2(3w + w) = 8w, and solving 8w = 64 results in w = 8. This problem connects geometry with algebra.
Given the linear function f(x) = 2x + 3, what is the value of x for which f(x) equals 11?
7
5
4
8
Setting 2x + 3 equal to 11, subtract 3 to get 2x = 8, and then dividing by 2 gives x = 4. This demonstrates function evaluation and inversion.
Solve the equation: 2(x + 4) - 3(x - 2) = x + 8.
4
3
2
6
Expanding both parts gives 2x + 8 - 3x + 6 which simplifies to -x + 14 = x + 8. Solving for x yields x = 3 after combining like terms and isolating the variable.
An online store sells notebooks for $5 each and pens for $1 each. If you must buy at least one of each and you have exactly $20, what is the maximum total number of items you can purchase?
15
17
14
16
To maximize the number of items, buy the minimum one notebook for $5 and spend the remaining $15 on pens at $1 each, totaling 15 pens. Combined, this gives 1 notebook + 15 pens = 16 items.
A taxi service charges a fare given by C = 3 + 2x, where x is the number of miles traveled. If the total fare is $15, how many miles were traveled?
5
7
8
6
Subtracting the fixed fee of $3 from the total fare gives 2x = 12, and dividing by 2 yields x = 6. This problem applies a linear cost model to find the unknown variable.
0
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Study Outcomes

  1. Understand the principles of linear equations and their solutions.
  2. Apply algebraic techniques to solve and simplify equation expressions.
  3. Analyze calculus-based optimization problems to identify maximum and minimum values.
  4. Synthesize algebraic and calculus methods to tackle real-world optimization challenges.
  5. Evaluate problem-solving strategies to improve exam readiness.

Linear Equations & Programming Cheat Sheet

  1. Master the Standard Form of a Linear Equation - The iconic y = mx + b packs two crucial details: m tells you how steep your line climbs, and b reveals where it crosses the y-axis. Once you recognize this pattern, graphing and tweaking equations becomes a breeze - no sweat, no fuss!
  2. Crack Systems of Equations - Whether you're eyeballing graphs, swapping variables, or adding and subtracting entire equations, each method gives you a toolkit for pinpointing where two lines collide. Think of graphing as a visual treasure map, substitution as your stealthy insider trick, and elimination as the bouncer knocking out unneeded terms.
  3. Translate Word Problems into Equations - Real-world puzzles - from splitting a bill to tracking speed and distance - suddenly bow down to your algebraic prowess once you nail the translation step. Jot down what you know, assign variables like a boss, and let the math do the talking.
  4. Get Cozy with Optimization Basics - Optimization is all about finding the "just right" point where a function hits its peak or valley. Picture a mountain hike: you want the tallest summit (maximum) or the deepest cave (minimum), and calculus gives you the GPS.
  5. Follow the Optimization Recipe - Step 1: Define your goal (the objective function). Step 2: Lay out your restrictions (constraints). Step 3: Rewrite so just one variable does all the heavy lifting, then find critical points and test them for ultimate victory (or minimal mishaps).
  6. Tackle Geometric Shape Challenges - Rectangles, boxes, cylinders - whatever the shape, you'll optimize area or surface by unleashing derivatives. It's like shaping clay with math: squeeze and stretch until you get the perfect dimensions.
  7. Spot and Interpret Critical Points - Whenever the derivative hits zero or goes undefined, you've found a VIP - the critical point. These are your prime suspects for local maxima or minima, so don't let any slip through the cracks!
  8. Use the Second Derivative Test - Think of the second derivative as a magnifying glass: positive means a cozy bowl (minimum), and negative means an upside-down bowl (maximum). It's your quick-check tool for figuring out exactly what kind of critical point you've got.
  9. Optimize Motion Problems - Distance, rate, and time join forces in these high-speed optimization quests. Define your travel time or velocity function, set up the right equations, and apply derivatives to cross the finish line in record time.
  10. Explore Optimization in Economics - Businesses want to maximize profits and minimize costs - your job is to frame revenue and expense functions, take derivatives, and find those sweet spots. It's real-world math that pays off, literally!
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