1.7 Systems of Linear Equations and Inequalities Word Problems

You will encounter problems on SAT Math where you will have to set up a system of linear equations and/or inequalities in order to solve the problem.  While the previous two articles walk you through solving a system of linear equations, and solving a system of inequalities, this article will show you how to set up a system of equations or inequalities based on what is given in the word problem on the SAT Math exam.

Here is a process for approaching SAT Math word problem:

• Quickly scan the problem.
  • Decide for yourself if you’re familiar with it enough to spend time on it and attempt to solve it, or
  • if it looks like a blob of unknown quantity, then just move on.
• If you decide to pursue solving the problem, go back to the beginning, look for the key information and note it down. The key information that you are looking for are all the things listed below:
  • what is being asked for,
  • the values that are given,
  • unknown quantities,
  • formulas or expressions (if there are any)
  • charts, graphs or tables.
• If you’re still confused with the concept or the information that is given to you, draw a sketch of the problem, remember, a picture is worth a thousand words!
• Start putting together the following:
  • how the key information pieces are related to one another,
  • how they can be put together to figure out what is being asked for (remember all the information to solve the problem is given to you, all you have to do is puzzle it together)
  • what is the independent variable which dictates the outcome of the dependent variable?
  • what is the dependent variable?
  • Note, the units will help you put the expression together or to come up with the solution.
    •  All the terms in the expression have to be expressed in the same units.
      • For example: you can’t add the number of hours to the dollar amounts in the expression, then what do you get?  A mess…
      • If you’re figuring out the cost of a job in ($), and you’re putting all the terms of the expression in ($), then you need to think about how to apply the information given to you that is not in ($) to figure out the needed dollar amounts.  Say, if you multiply the dollar amounts produced from the hours worked, you will get ($) earned, etc.
    • The units can help you figure out what operation to perform.
      • You can figure out many hours someone worked based on the units of other parameters given.  If you were given the total amount of money they made ($) and their hourly rate ($/hr), you can see that if you divide ($) by the rate ($/hr), you will get hours.  In which case this is correct.
• Solve!  And remember, all the information that you need to produce the answer is given to you in the problem.
• If you can’t solve the problem after you spent good amount of time on it, rule out the least probable answers and pick one of the most likely answers.  Then move onto the next problem.

So without further due, let’s solve some SAT Math system of linear equations/inequalities problems!

Example Problem

Each day a delivery truck goes 65 miles per hour (mph) on the highway and 40 mph on the city streets.  The maximum distance for the truck to cover every days is 100 miles in 7 hours or less.  If the delivery truck drives a miles on the highway and b miles on the city streets, write a system of inequalities that represents delivery truck’s daily goal.

First, identify all the key pieces of information:

Speed on highway – 65 mph

Speed on city streets – 40 mph

Maximum distance to cover – 100 miles

Maximum driving hours – 7 hours

Miles on the highway – a miles

Miles on the city streets – b miles

Find: system of inequalities

Second, just by reading this, we know that two terms have to be smaller or equal to 100 and the expression for that has to be in miles.  (since the units for all the terms in the equation have to be the same).  What parameters can we use to put together this inequality?

So it will be 100 ≥ …

What is given to us that can help us add the terms to this inequality? We have the distances that the truck drives – a miles on the highway and b miles on the city streets.

Can we add a and b together to make total distance per day driven?

That seems logical, but remember that instead of the equal sign, we will put less than or equal to 100.  The problem doesn’t state that the truck has to drive for 100 miles per day exactly, it states that the maximum distance for the truck to cover is 100 miles.

So the sum of a and b has to be less than or equal to 100.

Therefore our first inequality is

a+b≤100

All the terms are expressed in miles, so this is one of the confirmations that the inequality is set up correctly.

Now we need to set up a second inequality.  Let’s find a second restriction that is given in the problem.

The truck has to drive for 7 hours or less.  So one side of the inequality will be 7≥ …

What about the other side of this inequality?

We know that the inequality’s units are hours, since 7 stands for 7 hours.  What other pieces of information haven’t we used?

We have the speed, which is in miles per hour and we haven’t yet used it.  Even though it’s not in hours, it has hours in it.

How can we use the speed to set up the inequality in hours?

From physics, we know that distance divided by speed gives us the time.  The units confirm that as well, if we take miles and divide them by miles/hour, we get hours.

So we can use the speed and the distance (a and b) to set up the inequality that will give us the hours.  Let’s do it.

a/65 + b/40 ≤ 7

So this is our second inequality.

We now have to write both inequalities together and we have the answer to the problem.

a+b≤100

a/65 + b/40 ≤ 7

Are you still feeling confused?  Do you need some help with inequalities?  Feel free to set up one-on-one session with me and I’ll be more than happy to help you.   I’m available to tutor SAT Math online or in person for Tampa, Brandon or Riverview students.  Happy studying!

Practice problems

Mia’s total for bread and cake was $18.77 before tax.  Bread was tax free but she paid 8% tax on the cake.  Her total after tax was $20.01.  How much did she spent on the bread?

If a lies within the solution set of the inequality shown above, what is the maximum possible value of a?

Still confused?  Schedule one-on-one session with me or use contact form above to contact me.