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# Systems of linear equations word problems | Lesson

## What are systems of linear equations word problems, and how frequently do they appear on the test?

Systems of linear equations word problems ask us to translate real-world scenarios into a system of two linear equations with two variables. Often, we'll also be asked to solve the system.
On your official SAT, you'll likely see 1 question that tests your ability to write and/or solve a system of linear equations based on a real-world scenario.
This lesson builds upon an understanding of the following skills:
You can learn anything. Let's do this!

## How do I solve systems of linear equations word problems?

### Systems of equations examples

Systems of equations with elimination: apples and orangesSee video transcript
Systems of equations with substitution: coinsSee video transcript

### How do I write systems of linear equations?

Word problems that require us to write systems of linear equations have two unknown quantities and two different ways to relate them.
This means we need to write two linear equations, and each contains the two unknowns as variables. The Understanding linear relationships lesson details how to translate verbal descriptions into equations.
After we write the equations, we can solve the system using our preferred method(s) for solving systems of linear equations covered in the Solving systems of linear equations lesson.
To solve a system of linear equations word problem:
1. Select variables to represent the unknown quantities.
2. Using the given information, write a system of two linear equations relating the two variables.
3. Solve the system of linear equations using either substitution or elimination.

#### Let's look at an example!

A theater production charges dollar sign, 21 for adult tickets and dollar sign, 15 for student tickets. If the production sold 102 tickets for its opening night and made dollar sign, 1, comma, 932 in ticket sales, how many of each type of tickets were sold?

### Try it!

Try: translate a scenario into equations
The player of a trivia game receives 100 points for each correct answer and loses 25 points for each incorrect answer. Leona answered a total of 30 questions and scored a total of 2, comma, 125 points.
Write an equation that relates the total number of questions Leona answered to C, the number of questions she answered correctly and I, the number of questions she answered incorrectly.
equals, 30
Leona received 100 points for each correct answer. Write an expression for the number of points she received from answering C questions correctly.
Because Leona loses 25 points for each incorrect answer, the quantity 25, I should be
Write an equation that models the total number of points Leona scored using C and I.
equals, 2, comma, 125

Practice: write a system of linear equations
At Beans & Bagels, a bagel costs dollar sign, 1 more than a cup of coffee. If 4 cups of coffee and 6 bagels cost dollar sign, 31, which of the following systems of equations can be used to determine the price of a bagel, b, and the price of a cup of coffee, c ?

Practice: solve a system of linear equations
A group of 41 people are going to a concert together in 7 vehicles. Some of the vehicles can hold 7 people each, and the rest can hold 5 people each. Assuming all the vehicles are filled to capacity, exactly how many of the vehicles can hold 7 people?

Practice: solve a system of linear equations
Mike and Sally are math teachers at the same high school. If Mike has 6 more years of teaching experience than Sally has, and the two of them have 16 years of teaching experience combined, how many years of teaching experience does Sally have?

## Want to join the conversation?

• I am very confused on the Try It! question. Can anyone explain why they did C+I=30 instead of C-I=30?
• The total number of questions is 30.
Each question is either answered correct C or incorrect I.
The sum of both should be 30 since each of the thirty questions will be either C or I, there is no other option.

What you have written is:
The number of correct answers (which is 30 or less) minus (so it becomes even less) the number of incorrect answers (at least zero) is always equal to 30.

When in doubt, try to translate these statement to their statements about the real world and it often clears up where you went wrong.
• In example 3, from where did 0.05 and 0.25 come?
I do not understand.
• how do i find the equation for really complicated word problems of systems of equations
(1 vote)
• By using little tricks to make sense of the complex word problems, and maybe breaking them up. The first step is recognizing that you've got a systems of equations problem. Then, sometimes the first sentence of the problem may give you one equation, and the second sentence may give you another piece.
Another thing you can do is pay attention to units. If the units of an unknown is maybe a number of people for example, and another quantity in the problem is a rate per person, and another is an amount, you would multiply the rate by the time to equal the amount. Maybe there's multiple rates for each scenario, and you would have to add the amount from each to get the total amount. That's a really abstract way of saying it, but basically watch for units because in an equation, the units have to balance on each side just like the numbers, so maybe that can clue you in to some equation that you can build from the tricky word problem.
And above all, always, is practice. Practice gets you exposed to the types of tricky word problems on the SAT, so you can find out the best ways to tackle them and put them in your head.
(1 vote)
• I hate khan academy its basically a rage game
(1 vote)
• In the second video,where Sal talks about nickels and quarters,how did he come up with values for the two even when we aren't given any values in the question?
(1 vote)
• What is difference between SAT and LCAT (LUMS Common Admission Test)? I mean,to what extent do they differ?