Problem 35 - Precalculus with Limits

Question

Solving a System of Equations Graphically In Exercises \(33-44,\) solve the system graphically. $$\left\\{\begin{array}{l}{x-3 y=-3} \\ {5 x+3 y=-6}\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is the point of intersection of the two lines on the graph.

Step 1: Transform the first equation into slope-intercept form

First, rearrange \(x - 3y = -3\) to give \(y = (1/3)x + 1\). It is achieved by adding \(3y\) to both sides and then dividing by \(3\). This results in the equation of the first line in slope-intercept form.

Step 2: Transform the second equation into slope-intercept form

Next, rearrange \(5x + 3y = -6\) to give \(y = -(5/3)x -2\). It is done by subtracting \(5x\) from both sides and then dividing by \(3\). This results in the equation of the second line in slope-intercept form.

Step 3: Graph the two lines

On the same set of axes, graph the lines represented by the two equations. The line from the first equation has a slope of \(1/3\) and a y-intercept of \(1\). The line from the second equation has a slope of \(-(5/3)\) and a y-intercept of \(-2\).

Step 4: Identify the intersection (solution)

The solution to the system of equations is the point where the two lines intersect. It could be determined visually or found by solving the two equations algebraically.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to express a linear equation. It is given by the formula: \[ y = mx + b \]. In this equation, \(m\) represents the slope, and \(b\) is the y-intercept. The slope is a measure of how steep a line is, or how much \(y\) changes for every change in \(x\). The y-intercept is the point where the line crosses the y-axis.

• Slope \(m\): Indicates the tilt of the line. A positive slope means the line rises, while a negative slope means it falls.
• Y-Intercept \(b\): The value of \(y\) when \(x = 0\). It's where the line touches the y-axis.

By converting an equation into slope-intercept form, you make graphing straightforward. This is because you can easily identify the slope and y-intercept. For example, in the equation \(y = (1/3)x + 1\), the slope is \(1/3\), and the y-intercept is \(1\). This indicates that as \(x\) increases by 1, \(y\) increases by \(1/3\).

Intersection of Lines

The intersection of lines refers to the point where two lines on a graph cross each other. This point is significant because it represents the solution to a system of equations when graphed.

• Graphical Representation: When two equations are plotted as lines, their intersection point is where their y-values are equal for the same x-value.
• Solution: This intersection point is the solution to the system. In other words, the values of \(x\) and \(y\) at the intersection satisfy both equations simultaneously.

Finding the intersection point helps verify solutions derived algebraically. For the given equations, after transforming them to slope-intercept form, the intersection can be pinpointed on the graph. It visually confirms that the solutions meet at a particular x and y coordinate.

Graphing Techniques

Graphing is a visual method of displaying mathematical equations that makes it easier to understand relationships between variables. Here are some basic techniques:

• Identify Intercepts: Start by plotting the y-intercept \(b\). For example, if \(b = 1\), place the first point at (0,1).
• Use the Slope: From the y-intercept, use the slope \(m\) to find another point. If the slope is \(1/3\), move up 1 unit and right 3 units from the y-intercept.
• Draw the Line: Connect the points with a straight edge to extend the line across the graph. Be sure it covers a reasonable range along both axes.

To graph different equations on the same axes, repeat these steps for each one. Watching where the lines intersect gives a clear, visual solution to the system. This intuitive approach enhances comprehension and allows quick identification of solution points on a graph.