# BM1.7 – Solving Systems of Equations

Chapter 1, Lesson 7

In this lesson students will:

• understand what is meant by a system of equations
• solve a system of equations using the substitution method
• solve a system of equations using the elimination method

## Systems of Linear Equations

#### An Introduction to Systems of Linear Equations

In this lesson we are going to cover algebraic equations in more detail. The main difference in this lesson is that we will be working with equations that contain two variables $x$ and $y$. The main goal of this lesson will be to solve a system of two equations. Solving a system of equations means finding numerical values for $x$ and $y$ that satisfy both equations at the same time.

Consider the following system of equations:

\begin{aligned} 2x + 3y &= 13\\ x + 4y &= 14. \end{aligned}

We say that $x=2$ and $y=3$ solve the system because if we were to substitute these values into equation one we get $2x+3y = 2(2)+3(3) = 13$, and if we substitute these values into equation two we get $x+4y = (2) + 4(3) = 14$. In other words, both equations are true. The values $x=1$ and $y=5$ would not solve the system because equation one would not work: $2x+3y=2(1)+3(5) = 17$, which is not $13$.

#### Solving a System Using Substitution

The real challenge of solving a system is finding the values for $x$ and $y$ if we are not given this information to start with. Luckily there are two well-known methods we can use to solve a system of equations. The first method of solving a system of equations is called substitution. Substitution is beneficial if the coefficient of one of the variables in your system is one. Substitution is less helpful if when isolating in step one, you obtain decimal numbers or fractions. The main steps of the substitution method are summarized below.

1. Choose one equation and isolate for either $x$ or $y$. This means you should have an expression of the form $x = \hdots$ or $y = \hdots$ once you have completed this step.
2. Substitute the expression you obtained in step one into the second equation. The second equation should now have only one variable.
3. Simplify and solve the equation obtained. You should obtain a numerical value for one of the variables in the system.
4. Substitute this numerical value back into the very first equation you worked with to find the numerical value of the second variable.

The video will cover the basic four-step process of solving a system using this method.

#### Solving a System Using Elimination

The second method of solving a system of equations is called elimination. Elimination is helpful when there are many different coefficients in front of the variables, or when we notice that two coefficients of one of our variables happens to be similar. The main steps of the elimination method are summarized below.

1. The first step is to look at all the coefficients and determine if there is a match in the $x$ variable or a match in the $y$ variable. If there is, proceed to step two. If there is not, then we must multiply our equations by appropriate numbers to ensure a match. More information on how this works will be in the video.
2. Once you have a match, we subtract one equation from the other equation. Be careful of your signs in this step, as you might need to work with integer numbers.
3. After the subtraction step, you should have a new equation with only one variable remaining. Solve the equation obtained to get a numerical value for one of the variables.
4. Substitute this numerical value back into one of the first two equations you worked with to find the numerical value of the second variable.

The video will cover the process of solving a system using this method.