Systems - Substitution Word Problems
Lesson Plans > Mathematics > Algebra > Systems of EquationsSlide Show
Problem Solving Strategies
- Identify what the problem is asking for.
- Identify unknown information, and assign variables.
- Identify known information.
- Do any formulas (perimeter, area, etc) apply?
- Combine knowns and unknowns to make equations.
- Solve the equations.
- Use the solved equations to find the answer.
- Check your answer.
My neighbor has both chickens and roosters. He has a total of 31 birds. The number of chickens is ten more than twice the number of roosters. How many chickens does he have?
The difference between two numbers is twice the smaller number. The smaller number is 10 less than the larger number. Find the smaller number.
The shortest side of a triangle has length 16 inches. The perimeter of the triangle is nineteen less than three times the longest side.The remaining side is eight inches shorter than the longest side. What are the side lengths of the triangle?
Bob is three times as old as Bart. Bob is also twenty-two years older than Bart will be in two years. How old is each?
Lesson Plan/Article
Systems - Substitution Word Problems
In this lesson we focus on how to solve a simple system of equations in which one variable can easily be written in terms of another variable. This is the simplest kind of system of equations to solve. In fact, many students will solve these kinds of problems without even thinking of it as a system of equations. For this lesson, we require students to use two variables (or more), making these problems a nice transition from single-variable problems to systems problems. Each of these systems is easily solved using substitution.
Slide 1: Discuss with students some of the strategies involved in solving algebra problems. Emphasize that students should write a variable for each unknown quantity, and clearly write what that variable represents. For example, S = Sally's age.
Slides 2 - 6: Each of these is a word problem which is set up nicely for solving using substitution. Notes to consider are shown below.
Difference System: Depending on how it is solved, this system will require students to properly place parentheses around the variable being substituted, and then properly distribute a negative through the parentheses. This problem can also be used to discuss being careful about choice of variables. The obvious choices for variables in this problem are L and S (for larger and smaller). However, students who write carelessly may mistake a lower case L for a one, and an S for a 5. Thus, a script or upper case L is a better choice. One student in my class selected "B" and "T" for "Big" and "Tiny," thus avoiding the issue altogether.
Sum of Two Fractions: This system can be solved by clearing the equations of fractions and then solving, or by carrying the fraction through the problem. Do whichever seems more suitable for your class, or show both ways of solving. Keep in mind that clearing the fractions causes it to no longer be a simple substitution problem.
Triangle Problem and Age Problem: In keeping with the idea that we specify a variable for each unknown, these will have more than two unknowns. The triangle problem has two unknown sides and the perimeter. Strictly speaking, the age problem also has three unkowns: Bob's age, Bart's age, and Bart's age two years from now. Students will hopefully see quickly that Bart's age in two years is 2 more than it is now. Bob and Bart's names were specifically chosen to force students to recognize that B can't be used as a variable for both, and another choice must be made for one of the boys.
Handouts/Worksheets
Problems
Chickens and Roosters
My neighbor has both chickens and roosters. He has a total of 31 birds. The number of chickens is ten more than twice the number of roosters. How many chickens does he have?
Difference System
The difference between two numbers is twice the smaller number. The smaller number is 10 less than the larger number. Find the smaller number.
Sum of Two Fractions
Triangle Perimeter
The shortest side of a triangle has length 16 inches. The perimeter of the triangle is nineteen less than three times the longest side.The remaining side is eight inches shorter than the longest side. What are the side lengths of the triangle?
Bob's age and Bart's Age
Bob is three times as old as Bart. Bob is also twenty-two years older than Bart will be in two years. How old is each?
Solutions
This content is for teachers only, and can only be accessed with a site subscription.Word Problem Worksheet
Instructions: Pick a variable for each unknown, and state what it represents. Find a way to write one variable in terms of the other. Write an equation and solve.
Example: Alfred is four years older than Tina. Together they are 36 years old. How old is Alfred?
Answer: t = Tina’s current age
a = Alfred’s age; a = t + 4
t + a=36; t + (t + 4) = 36; t = 16; a = 20
- Jonas has three more nickels than dimes. He has 41 coins in all. How many are dimes?
- The main story of a novel has 120 times as many pages as the prologue. There are 363 pages in all. How many pages are in the main story?
- The number of nails in the bucket is 50 less than twice the number of screws. Together, there are 400 fasteners in the bucket. How many of each are there?
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- Yvonne did three times as many jumping-jacks as Cora did. Together they did 100 jumping-jacks. How many did Cora do?
- The number of crayons in a box is forty less than twice the number of markers. Together, there are 140 coloring tools. How many are crayons?
- There are twenty-two more chickens than cows. The number of cows and chickens combined is 98. How many are chickens?
- A car travels travels from City A to City B, and then from City B to City C. City C is 37 miles further away from City B than City A is from City B. The entire journey is 279 miles. How far is it from City B to City C?
- A rectangle’s width is 12 feet longer than its length. Its perimeter is 80 feet. What are the dimensions of the rectangle?
- The two longer sides of a triangle are 12 units longer than the shortest side and 7 units longer than the shortest side. The triangle’s perimeter is 58 units. What are the triangle’s side lengths?
Word Problem Worksheet: Answer Key
This content is for teachers only, and can only be accessed with a site subscription.
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