Problem 28 - Elementary Algebra

Question

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing. $$ \left\\{\begin{array}{l} -x+3 y=3 \\ x+3 y=3 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system of equations is \((0, 1)\).

Step 1 - Rewrite each equation in slope-intercept form

First, rewrite each equation in the form of \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. The given system is: \[-x + 3y = 3\] and \[x + 3y = 3\].

Step 2 - Solve the first equation for y

Solve \[-x + 3y = 3\] for \( y \): \[\begin{array}{rl} 3y & = x + 3 \ \ y & = \frac{1}{3}x + 1 \end{array}\]

Step 3 - Solve the second equation for y

Solve \[x + 3y = 3\] for \( y \): \[\begin{array}{rl} 3y & = -x + 3 \ \ y & = -\frac{1}{3}x + 1 \end{array}\]

Step 4 - Plot the equations on a graph

Now plot the lines \[ y = \frac{1}{3}x + 1 \] and \[ y = -\frac{1}{3}x + 1 \] on a graph. These are straight lines, and they will intersect at some point if there is a unique solution.

Step 5 - Find the intersection point

Determine the point where the two lines intersect. This is the solution to the system of equations. After plotting, observe that both lines intersect at \((0, 1)\).

Key Concepts

graphing linear equations

Graphing linear equations is a fundamental skill in algebra. It allows you to visually represent the relationship between variables and see their behavior on a Cartesian plane. For the given system, we start by understanding that each equation represents a line.
To plot these lines, convert the equations into slope-intercept form:

• First, rearrange them as y = mx + b, to easily identify the slope and the y-intercept.
• When you graph them, the x-axis and y-axis intersect, creating a grid where you can plot points.

In this case, we have two lines to graph.

slope-intercept form

The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. Using this form makes it easy to graph the equation:
Let's start with our equations:

• For \[-x + 3y = 3\], solving gives us y = \(\frac{1}{3}x + 1\).
• Similarly, for \[x + 3y = 3\], solving gives y = \(-\frac{1}{3}x + 1\).

Here, the slope \(m\) determines the steepness of the line, and \(b\) indicates where it crosses the y-axis.
This form is especially useful for quickly plotting and understanding the behavior of linear functions.

intersection of lines

The intersection of lines on a graph represents the solution to the system of equations. Here's how you find it:

• By plotting both equations on the graph,
• observe where they meet.

For \(y = \frac{1}{3}x + 1\) and \(y = -\frac{1}{3}x + 1\), as they cross at the same point, that point is \(0, 1\).
This intersection point \(0, 1\) is the one and unique solution to the system.
Understanding intersections helps visualize solutions and identify whether a system has one solution, infinitely many, or none.

system of equations

A system of equations consists of two or more equations with the same variables. Solving these systems means finding the values of the variables that satisfy all equations simultaneously. To recap:

• Start by converting the equations into the same form, such as slope-intercept form.
• Then, graph each equation on the same set of axes.
• Lastly, determine where the lines intersect, which represents the solution.

For instance, in our given problem: \[-x+3y=3\] and \ x+3y=3 \, after graphed, intersect at \(0, 1\).
Graphing is one of the several methods to solve systems of equations, alongside substitution and elimination.
Each method provides different insights and techniques to handle various algebraic challenges.