# Systems of Linear Equations and Word Problems

Note that we saw how to solve linear inequalities here in the Coordinate System and Graphing Lines section. Note also that we solve Algebra Word Problems without Systems here, and we solve systems using matrices in the Matrices and Solving Systems with Matrices section here.

## Introduction to Systems

“Systems of equations” just means that we are dealing with more than one equation and variable. So far, we’ve basically just played around with the equation for a line, which is $y=mx+b$. Let’s say we have the following situation:

You’re going to the mall with your friends and you have $200 to spend from your recent birthday money. You discover a store that has all jeans for$25 and all dresses for $50. You really, really want to take home 6 items of clothing because you “need” that many new things. How many pairs of jeans and how many dresses you can buy so you use the whole$200 (tax not included)?

Now, you can always do “guess and check” to see what would work, but you might as well use algebra! It’s much better to learn the algebra way, because even though this problem is fairly simple to solve, the algebra way will let you solve any algebra problem – even the really complicated ones.

The first trick in problems like this is to figure out what we want to know. This will help us decide what variables (unknowns) to use. What we want to know is how many pairs of jeans we want to buy (let’s say “$j$”) and how many dresses we want to buy (let’s say “$d$”). Always write down what your variables will be:

Let $j=$ the number of jeans you will buy
Let $d=$ the number of dresses you’ll buy

Like we did before, let’s translate word-for-word from math to English:

Now we have the 2 equations as shown below. Notice that the $j$ variable is just like the $x$ variable and the $d$ variable is just like the $y$. It’s easier to put in $j$ and $d$ so we can remember what they stand for when we get the answers.

This is what we call a system, since we have to solve for more than one variable – we have to solve for 2 here. The cool thing is to solve for 2 variables, you typically need 2 equations, to solve for 3 variables, you need 3 equations, and so on. That’s easy to remember, right?

We need to get an answer that works in both equations; this is what we’re doing when we’re solving; this is called solving simultaneous systems, or solving system simultaneously. There are several ways to solve systems; we’ll talk about graphing first.

## Solving Systems by Graphing

Remember that when you graph a line, you see all the different coordinates (or $x/y$ combinations) that make the equation work. In systems, you have to make both equations work, so the intersection of the two lines shows the point that fits both equations (assuming the lines do in fact intersect; we’ll talk about that later). The points of intersections satisfy both equations simultaneously.

Put these equations into the $y=mx+b$ ($d=mj+b$) format, by solving for the $d$ (which is like the $y$):

$\displaystyle \begin{array}{c}j+d=6;\text{ }\,\text{ }\,\text{solve for }d:\text{ }d=-j+6\text{ }\\25j+50d=200;\text{ }\,\,\text{solve for }d:\text{ }\\d=\displaystyle \frac{{200-25j}}{{50}}=-\displaystyle \frac{1}{2}j+4\end{array}$

Now graph both lines:

We can also use our graphing calculator to solve the systems of equations:

Note that with non-linear equations, there will most likely be more than one intersection; an example of how to get more than one solution via the Graphing Calculator can be found in the Exponents and Radicals in Algebra section. Also, there are some examples of systems of inequality here in the Coordinate System and Graphing Lines section.

## Solving Systems with Substitution

Substitution is the favorite way to solve for many students! It involves exactly what it says: substituting one variable in another equation so that you only have one variable in that equation. Here is the same problem:

You’re going to the mall with your friends and you have $200 to spend from your recent birthday money. You discover a store that has all jeans for$25 and all dresses for $50. You really, really want to take home 6 items of clothing because you “need” that many new things. How many pairs of jeans and how many dresses you can buy so you use the whole$200 (tax not included)?

Below are our two equations, and let’s solve for “$d$” in terms of “$j$” in the first equation. Then, let’s substitute what we got for “$d$” into the next equation. Even though it doesn’t matter which equation you start with, remember to always pick the “easiest” equation first (one that we can easily solve for a variable) to get a variable by itself.

We could buy 4 pairs of jeans and 2 dresses. Note that we could have also solved for “$j$” first; it really doesn’t matter. You’ll want to pick the variable that’s most easily solved for. Let’s try another substitution problem that’s a little bit different:

## Solving Systems with Linear Combination or Elimination

Probably the most useful way to solve systems is using linear combination, or linear elimination. The reason it’s most useful is that usually in real life we don’t have one variable in terms of another (in other words, a “$y=$” situation).

The main purpose of the linear combination method is to add or subtract the equations so that one variable is eliminated. We can add, subtract, or multiply both sides of equations by the same numbers – let’s use real numbers as shown below. We are using the Additive Property of Equality, Subtraction Property of Equality, Multiplicative Property of Equality, and/or Division Property of Equality that we saw here in the Types of Numbers and Algebraic Properties section:

If we have a set of 2 equations with 2 unknowns, for example, we can manipulate them by adding, multiplying or subtracting (we usually prefer adding) so that we get one equation with one variable. Let’s use our previous problem:

We could buy 4 pairs of jeans and 2 dresses.

Here’s another example:

## Types of equations

In the example above, we found one unique solution to the set of equations. Sometimes, however, for a set of equations, there are no solutions (when lines are parallel) or an infinite number of solutions or infinitely many solutions (when the two lines are actually the same line, and one is just a “multiple” of the other).

When there is at least one solution, the equations are consistent equations, since they have a solution. When there is only one solution, the system is called independent, since they cross at only one point. When equations have infinite solutions, they are the same equation, are consistent, and are called dependent or coincident (think of one just sitting on top of the other).

When equations have no solutions, they are called inconsistent equations, since we can never get a solution

Here are graphs of inconsistent and dependent equations that were created on a graphing calculator:

## Systems with Three Equations

Let’s get a little more complicated with systems; in real life, we rarely just have two unknowns to solve for.

### Geometry Word Problem:

Many times, we’ll have a geometry problem as an algebra word problem; these might involve perimeter, area, or sometimes angle measurements (so don’t forget these things!). Let’s do one involving angle measurements.

See – these are getting easier! Here’s one that’s a little tricky though:

### Work Problem:

Let’s do a “work problem” that is typically seen when studying Rational Equations (fraction with variables in them) and can be found here in the Rational Functions, Equations and Inequalities section.

Note that there’s also a simpler version of this problem here in the Direct, Inverse, Joint and Combined Variation section.

### Three Variable Word Problem:

Let’s do one more with three equations and three unknowns:

### The “Candy” Problem

Sometimes we get lucky and can solve a system of equations where we have more unknowns (variables) then equations. (Actually, I think it’s not so much luck, but having good problem writers!) Here’s one like that:

There are more Systems Word Problems in the Matrices and Solving Systems with Matrices section, Linear Programming section, and Right Triangle Trigonometry section.

Understand these problems, and practice, practice, practice!

For Practice: Use the Mathway widget below to try a Systems of Equations problem. Click on Submit (the blue arrow to the right of the problem) and click on Solve by Substitution or Solve by Addition/Elimination to see the answer.

You can also type in your own problem, or click on the three dots in the upper right hand corner and click on “Examples” to drill down by topic.

If you click on Tap to view steps, or Click Here, you can register at Mathway for a free trial, and then upgrade to a paid subscription at any time (to get any type of math problem solved!).

On to Scatter Plots, Correlation, and Regression – you are ready!

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