An architect designing a home may have restrictions with both the area and the perimeter of the windows because of energy and structural concerns. The length and width chosen for each window would have to satisfy two equations: one for the area, and the other for the perimeter. Similarly, a banker may have a fixed amount of money to put into two investment funds. A restaurant owner may want to increase profits, but in order to do that, he will need to hire more staff, which would increase his expenses as well. The list can go on and on regarding situations where there are multiple things taken into consideration. Mathematically, if you're trying to solve a problem that has multiple equations, we call this a System of Equations.
A system of equations is when you take two or more equations and try to solve them simultaneously (at the same time). These types of models are frequently used in business to determine profit. The profit a business makes is the difference between the cost and revenue. If the cost is higher than the revenue, a company loses money. But if the revenue is higher than the cost, then the company will make money. But, before we get into all of that, we will look at how to algebraically verify a solution to a system of equations.
Determine if the ordered pair $(2,-1)$ is a solution to the system:
If this point is a solution to this system, then the coordinate will satisfy both equations by making both equations true statements.
Equation 1: $3x - y = 7$
\begin{align*} 3(2)-(-1)&=7\\ 6+1&=7\\ 7&=7\\ &\text{True} \end{align*}Equation 2: $x - 2y = 4$
\begin{align*} 2-2(-1)&=4\\ 2+2&=4\\ 4&=4\\ &\text{True} \end{align*}It can be seen that when you substitute $(2,-1)$ into each equation and simplify, you end up with two true statements, verifying that $(2,-1)$ is a solution to the system algebraically.
Determine if the points $(1,-3)$ and $(0,0)$ are ordered pairs of the system:
As we learned in the previous section, the graph of a linear equation is a straight line. Each point on the line is a solution to that particular equation. For a system of equations, you'd graph each line independently of each other. The point(s) where those equations all intersect would be the solution(s) to the system of equations.
When it comes to systems of equations, there are three options for solution types:
| One Solution | No Solution | Infinite Solutions |
|---|---|---|
|
The equations only intersect at one singular point. |
The equations never intersect and are parallel. |
The equations are the same and overlap each other. |
Before we get into an example, let's talk about the slope-intercept method for graphing linear equations. In the previous unit, we revisited graphing by using an $x/y$ table. With that method, you substitute values in for $x$ to find values of $y$ to plot the ordered pair $(x,y)$. That method can entail a lot of work. While you can always use that method, we will venture into the slope-intercept method. An equation in the form of $y=mx+b$ is an equation in slope-intercept form. The variable $m$ represents the slope, or "rate of change", and the variable $b$ represents the y-intercept (point $(0,b)$).
To use this method, follow these steps:
Graph the equation $-2x+y=4$.
The first step we need to take is to solve our equation for $y$.
\begin{align*} -2x+y&=4\\ y&=2x+4 \end{align*}Next, we need to identify the slope $m=2=\dfrac{2}{1}$. This means we'd rise 2 units and run 1 unit. Our y-intercept is next in the process, which is the point $(0,4)$. To create this graph, we'd first plot the y-intercept, then proceed with using the slope to plot the second point as outlined in the graph below.
Now that we've outlined that method for graphing equations, let's apply it to a system of equations problem.
Graphically find the solution(s) to the system:
To find the solution, we need to graphically plot each line and then find the point(s) at which they intersect. To plot each line, we are going to use the slope-intercept method demonstrated previously. To do that, we need to solve each equation for $y$.
Equation 1:
\begin{align*} 2x+y&=7\\ y&=-2x+7 \end{align*}Equation 2:
\begin{align*} x-2y&=6\\ -2y&=-x+6\\ \frac{-2y}{-2}&=\frac{-x+6}{-2}\\ y&=\frac{1}{2}x-3 \end{align*}Now we are ready to plot the two equations $y=-2x+7$ and $y=\frac{1}{2}x-3$. Next, we will need to identify the slope and y-intercept for each equation.
Equation: $y=-2x+7$
Slope: $m=-2=\dfrac{-2}{1}$
y-intercept: $(0,7)$
Equation: $y=\frac{1}{2}x-3$
Slope: $m=\dfrac{1}{2}$
y-intercept: $(0,-3)$
For reference, the equation $y=-2x+7$ will be plotted in blue and the equation $y=\frac{1}{2}x-3$ will be plotted in green.
It can be seen from the graph that the two lines intersect at the point $(4,-1)$, which is denoted with a red point on the graph. This is the solution to this system of equations.
Solve the system of equations by graphing:
At the beginning of this section, we spoke about the different real-world applications a system of equations can take on. The majority of the time, the actual equations will not be provided. Instead, a verbal argument will be posed, and from that verbal argument, you'll need to write the equations. Let's look at an application problem.
A business wants to manufacture bicycle frames. To be cautious, before they start making the frames, they want to make a model for profit and one for the materials and labor costs. Their accountant determines that it will cost them a start-up fee of \$ 35,000 with a price of \$0.85 per frame produced. They have also decided that they will sell each frame for \$1.55.
1. Find the Cost Equation
From the problem, "Their accountant determines that it will cost them a start-up fee of \$35,000 with a price of \$0.85 per frame produced". What this tells us is that we have a starting point, or y-intercept, of \$35,000, and for each frame sold, it will cost an additional \$0.85. This would produce us the equation:
2. Find the Revenue Equation
Once again looking back at the problem, we see that the company has decided to sell each frame for \$1.55, which would produce us the equation:
In both cases, $y$ is the total amount of money to produce $x$ bicycle frames.
3. Identify the break-even point
Recall the break-even point is where cost and revenue are equal. Essentially, it is asking us to solve the system of equations:
Next we will graph each of those equations. To do that, we once again will employ one of our graphing methods. Also to put this graph on a scale that works, the $y$-axis needs to be scaled by 10,000 and the $x$-axis needs to be scaled by 10,000. What that means is the point $(5,0)$ represents $5 \times 10,000=50,000$ on the $x$-axis, and the point $(0,3.5)$ represents $3.5 \times 10,000=35,000$ on the $y$-axis. This allows us to plot larger numbers on a smaller scale.
What this graph tells us is that the company will break-even (cost = revenue) when the company produces $5 \times 10,000=50,000$ bicycle frames. At that point, the cost and the revenue is $7.5 \times 10,000=\$75,000$.
Important Note: When using models, it is extremely important to understand what each variable represents and how they are being scaled. It is important to understand when axes are being scaled by different numbers, this allows the reader to interpret what they are looking at better. For example, in this graph, while the axes were scaled by the same number, it is important for the reader to understand that one unit on the graph actually represents 10,000 units.