Problem 122 - Introductory Algebra for College Students

Question

Solve the system by graphing: \(\left\\{\begin{array}{rr}2 x-y= & -4 \\ x-3 y= & 3\end{array}\right.\) (Section 4.1, Example 2)

Step-by-Step Solution

Answer

The solution to the system of equations is the point (3, 2).

Step 1: Rewrite the equations in slope-intercept form

First, both equations need to be rewritten in slope-intercept form, which looks like \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept. For the first equation, solve for \(y\) to obtain the equation \(y = 2x + 4\). Similarly, for the second equation, solve for \(y\) to obtain the equation \(y = \frac{1}{3}x - 1\).

Step 2: Graph the equations

Next, graph both equations on the same set of axes. For the first equation \(y = 2x + 4\), start at the y-intercept of 4 on the y-axis and then use the slope to plot additional points (the slope, 2, indicates to go up 2 units and over 1 unit). Repeat the same for the other equation \(y = \frac{1}{3}x - 1\). Plot the y-intercept at -1 and then add points using the slope of \(\frac{1}{3}\), which means go up 1 unit and over 3 units.

Step 3: Identify the point of intersection

Finally, identify the point where the two lines intersect. This point is the solution to the system of equations. After plotting the graph for the given equations, it can be found that the point of intersection is (3, 2).

Key Concepts

Slope-Intercept Form

When we look at a graph of a linear equation, the slope-intercept form is like a secret map that tells us exactly where to start our journey (the y-intercept) and how steep our path should be (the slope). In mathematical terms, the slope-intercept form is written as \( y = mx + b \). Here, \( m \) stands for the slope, kind of like the incline of a hill, and \( b \) is where we begin our graphing adventure on the y-axis.

In the example given in the exercise, we had to put on our algebra hats to transform \( 2x - y = -4 \) into a friendlier slope-intercept form. We did a bit of math magic, moving things around until we got \( y = 2x + 4 \). That's when the map became clear: we start at \( b = 4 \), and with every step right along the x-axis, we climb up two steps (\( m = 2 \)). Hooray for the treasure that is clarity!

When we graph, understanding the slope-intercept form is crucial. It simplifies our work and points us directly to where we need to graph without any guesswork. It’s like knowing the exact route to a hidden treasure chest!

Graphing Linear Equations

Graphing linear equations is similar to plotting a course on a treasure map. Each equation gives us a line, which is like a trail that leads to our treasure—the solution! To graph a line, we only need two things: the slope and the y-intercept both of which we find from the slope-intercept form.

For this treasure hunt, we use the slope-intercept form of our equations as our guide. Let's take \( y = 2x + 4 \) for an example. We start our journey at the y-intercept, which is the treasure's starting point located at \( y = 4 \) on the y-axis. From there, we follow the slope's instructions: for the '2' in \( m = 2 \), we move up two spaces (the rise) and go one space to the right (the run) from our starting point.

Once we mark enough points following our slope's directions, we draw a straight line through them, paving our way to possible treasure locations. The exercise transforms our graph into a treasure map, where 'x' marks the spot. Students that master graphing can confidently set sail on the mathematical seas, knowing they have the skills to find the treasure!

Point of Intersection

The point of intersection is like the X that marks the spot on a treasure map—the precise location where the treasure lies. In terms of algebra, it's the magical spot where two lines cross paths. This point is the solution to our system of equations, because it's the set of coordinates that satisfies both equations simultaneously. Essentially, this is the 'Eureka!' moment for every math treasure hunter.

In our example, we plotted two lines from our equations: \( y = 2x + 4 \) and \( y = \frac{1}{3}x - 1 \). It's like following two different treasure maps that lead to the same spot. Once we've graphed our lines, we look for where they cross, and that's our point of intersection. For this particular quest, the lines intersect at the coordinates (3, 2).

Whenever you find the point where two lines meet on a graph, you've struck gold, algebraically speaking. So remember, keep a keen eye out for the point of intersection; it’s where the treasure of certainty lies in the world of equations!