Understand the definition of a solution to a simultaneous system of linear equations.
Describe the three configurations for a system of two linear equations.
Determine the configuration of a system of two linear equations.
Solve systems of linear equations using substitution.
In the previous sections, we were looking at lines in isolation of each other. In this section, we’re going to look at the ways that two lines can interact with each other.
A solution to a linear equation is a pair of values \((x, y)\) that make the equation true. When we talk about a simultaneous system of linear equations, we are looking for a pair of values \((x, y)\) that satisfy all of the equations at the same time. This is probably best understood graphically.
The solution of a single linear equation can be represented as a line. Every point on this line can be plugged into that equation and yield a true result. If we add in a second linear equation, we get a different line. These are represented as the two separate graphs below.
When we talk about solutions of simultaneous equations, we want to find points that satisfy both equations at the same time. In other words, we’re looking for points that are on both lines, and so we can merge the two images into one.
Definition15.1.System of Simultaneous Linear Equations.
A system of simultaneous linear equations is a collection of linear equations that are to be solved at the same time, if possible. A solution of a system of simultaneous equations is a point \((x, y)\) that is a solution of all the equations.
Activity15.1.Verifying Solutions of Simultaneous Linear Systems.
We are going to focus on systems of two linear equations with two variables. Consider the following system:
\begin{equation*}
\left\{
\begin{array}{rcrcr}
x \amp + \amp y \amp = \amp 2 \\
x \amp - \amp y \amp = \amp 4
\end{array}
\right.
\end{equation*}
As noted in the definition, a solution must solve all the equations. This means that even though \((1, 1)\) is a solution of \(x + y = 2\) and \((2, -2)\) is a solution of \(x - y = 4\text{,}\) neither nor would be a solution to the system because when we plug in the values, we don’t solve both.
Try it!
Show that \((3, -1)\) is a solution to the system of equations by direct substitution.
Solution.
When thinking about different possibilities for solutions, we need to think about the configurations we can get when we graph the two lines together. It turns out that there are three possibilities:
In the first case, we can see that there are no solutions because there do not exist any points that lie on both lines at the same time. In the second case, we can see that every point on one line is also a point on the other line, so there are infinitely many solutions. In the last case, we can see that there is exactly one point that lies on both lines.
Activity15.2.Systems in Slope-Intercept Form.
If the equations are given in slope-intercept form, it’s possible to immediately identify which situation you are in, and it’s also possible to use substitution to solve the equations. Suppose that you have the following system of equations, where and are all constants.
\begin{equation*}
\left\{
\begin{array}{rcrcr}
y \amp = \amp m_1 x \amp + \amp b_1 \\
y \amp = \amp m_2 x \amp + \amp b_2
\end{array}
\right.
\end{equation*}
We can match these up with the images above.
Two different parallel lines: Same slope, different intercept
Two overlapping lines: Same slope, same intercept
Two non-parallel lines: Different slopes
If the lines are not parallel, you can find the solution by substituting one equation into the other.
Try it!
Describe the configuration of the following system of equations. If they intersect at a single point, determine the coordinates of that point.
This is an example of solving an equation by substitution. The idea of a substitution here is the same as we used earlier, where we simply replace a symbol in an equation with some other collection of symbols, which then reduces the equation to a single variable equation that we can solve.
Activity15.3.Solving a System of Linear Equations.
In general, the equations will not be given to you in slope-intercept form. This means that you will have to choose which variable to solve for and from which equation. There are no rules for this. You can sometimes avoid fractions, but you often can’t. Here is the same system of equations solved two ways. Each column represents a different approach, but notice that they end up in the same place.
Try it!
Solve the system of equations above two more times by solving for \(x\) and \(y\) from the second equation.
Solution.
If the system of equations involves two lines with the same slope, but the equations are not written in slope-intercept form, you may not immediately recognize that they have the same slope. However, in the process of working through the algebra, you will end up with all of the variable terms canceling out. If that happens, the equation that you’re left with will tell you whether the lines overlap or not. If you get a valid mathematical equation, such as \(0 = 0\text{,}\) then the two lines overlap. If you end up with an invalid mathematical equation, such as \(2 = 5\text{,}\) then the lines do not overlap.
Subsection15.1.1Brief Fraction Review
We will take a deeper dive into fractions in a later section. For now, we’re just going to review the basic mechanics of fractions by looking at some examples.
For addition and subtraction of fractions, you need to use a common denominator. When rewriting a fraction with a different denominator, you must multiply both the numerator and denominator by the same value.
For multiplication of fractions, you need to multiply straight across. If you are multiplying by an integer, you can put that integer over a denominator of 1. If you remember how to reduce before multiplying, that’s fine. If you don’t, you can just reduce after multiplying. The problems in this section will not involve large enough numbers for that to be important. Make sure to reduce your fractions when you finish.
Suppose that \(m_1 \neq m_2\text{.}\) Then the following system of equations intersect at a single point. Determine the coordinates of that point.
\begin{equation*}
\left\{ \begin{array} {rl}
y \amp = m_1 x + b_1 \\
y \amp = m_2 x + b_2
\end{array} \right.
\end{equation*}
2.
Suppose that the following system of equations intersect at a single point. Determine the coordinates of that point.
\begin{equation*}
\left\{ \begin{array} {rcrcr}
ax \amp + \amp by \amp = \amp c \\
dx \amp + \amp ey \amp = \amp f
\end{array} \right.
\end{equation*}
Section15.3Deliberate Practice: Solving by Substitution
Algebra is a skill, which means it requires practice to become proficient. But it will take more than rote repetition to get there. Deliberate practice is the thoughtful repetition of a task. For each of these sections, you will be given a list of specific skills or ideas to focus on as you practice thinking through the problems.
Focus on these skills:
Write the original system of equations.
Use sentences to organize the steps of your work, including a concluding statement.
Instructions: Determine the configuration of the system of equations. If they intersect at a single point, determine the coordinates of that point using the method of substitution.
This section is primarily an exercise in your core algebra skills. Solve a linear equation for a variable, make a substitution, and then solve for a variable. These are all things that we’ve already done previously. What’s new about it is that those skills are being put together in a brand new way.
It is a common theme in mathematics that we build new ideas on the backs of old ideas. This is also where we can start to identify weaknesses in our understanding, as there may be one step in particular that appears to be more difficult than the others. For example, many students do fine with these problems when all of the numbers are integers, but as soon as fractions start to appear they start to struggle.
This is one of the challenges of the way the field of mathematics is built out. A lot of ideas are built on the foundation of your basic skills. And so even as you move forward into higher levels of math courses, you need to make sure you take the time to go backward to fill in the gaps that you have. And that’s one of the main goals of this book.
As you continue to fill in those gaps, you will start to discover that you’re able to make connections between ideas more easily, that you’re able to use the tools that you have already developed more efficiently, and that you’re able to see how to apply your ideas more effectively. In the end, those are the markers of mature mathematical thinking.
