Problem 29 - Algebra 1

Question

Use the graphing method to solve the linear system and tell how many solutions the system has. $$ \begin{aligned}&\frac{3}{4} x+\frac{1}{2} y=10\\\&-\frac{3}{2} x-y=4\end{aligned} $$

Step-by-Step Solution

Answer

The system of equations has one solution, which is (4,8).

Step 1: Rewrite the Equations in Slope-Intercept Form

First, transform each equation into the slope-intercept form \(y = mx + b\), where m is the slope, and b is the y-intercept. The first equation becomes \(y = - \frac{3}{2}x + 20\) and the second equation simplifies to \(y = \frac{3}{2}x - 4\).

Step 2: Graph the Equations

Next, we will graph these two lines. The first line (-3/2, 20) connects points at (0,20) and (13.33,0). The second line (3/2, -4) connects points at (0,-4) and (2.67,0). If done correctly, the two lines should intersect corresponding to the solution of the system.

Step 3: Identify the Point of Intersection

After drawing, it can be observed that the lines intersect at the point (4,8). This means that \(x = 4\) and \(y = 8\) is the solution of the system.

Step 4: Determine the Number of Solutions

As there exists a unique point of intersection, it can be concluded that the system of equations has one solution.

Key Concepts

Linear Systems

A linear system consists of two or more linear equations that share the same set of variables. When solving a linear system, we aim to find values for the variables that satisfy all equations in the system simultaneously. In our exercise, we are given two equations:

• \( \frac{3}{4}x + \frac{1}{2}y = 10 \)
• \(-\frac{3}{2}x - y = 4 \)

To solve this system graphically, we need to find the point where the graphs of these equations intersect. This point of intersection represents the set of values for \(x\) and \(y\) that make both equations true at the same time. Graphical methods help visualize the solutions by displaying where the two lines cross on a coordinate plane. Each line represents one equation, and their intersection shows the solution to the system. If the lines intersect at exactly one point, the system has a unique solution.

Slope-Intercept Form

The slope-intercept form is a way to express linear equations that makes graphing them straightforward. It is written as \(y = mx + b\), where:

• \(m\) is the slope of the line, indicating its steepness and direction.
• \(b\) is the y-intercept, the point where the line crosses the y-axis.

For this exercise, we transformed the given equations into slope-intercept form:

• The first equation becomes \(y = -\frac{3}{2}x + 20\), with a slope of \(-\frac{3}{2}\) and a y-intercept of 20.
• The second equation simplifies to \(y = \frac{3}{2}x - 4\), with a slope of \(\frac{3}{2}\) and a y-intercept of -4.

This form is useful for quickly plotting lines on a graph. You can start by plotting the y-intercept and then using the slope to find other points on the line. A positive slope means the line ascends from left to right, while a negative slope indicates it descends.

Point of Intersection

The point of intersection is crucial in solving a system of linear equations graphically. This is the point where two lines on a graph meet, representing the solution to the system.
In our solution, we plotted the two lines derived from the equations:

• First line: \(y = -\frac{3}{2}x + 20\)
• Second line: \(y = \frac{3}{2}x - 4\)

By graphing these lines, we observed that they intersect at the point \((4, 8)\). This tells us that the values \(x = 4\) and \(y = 8\) satisfy both equations, hence solving the system.
Such a graphical approach is beneficial especially when dealing with visual learners, as it gives an immediate visual indication of where the solution lies. The uniqueness of this point means there is only one solution for the system, reinforcing the idea that the two lines intersect at a single distinctive point on the graph.