Problem 33 - Algebra 2

Question

Solve each system of equations by graphing. \(\frac{2}{3} x+y=-3\) \(y-\frac{1}{3} x=6\)

Step-by-Step Solution

Answer

The solution to the system of equations is \((-9, 3)\).

Step 1: Rewrite each equation in slope-intercept form

The slope-intercept form of a linear equation is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Let's rewrite both equations in this form.\For the first equation \( \frac{2}{3}x + y = -3 \), subtract \( \frac{2}{3}x \) from both sides to get: \\[ y = -\frac{2}{3}x - 3 \]\This equation is now in slope-intercept form \( y = mx + b \), with slope \( m = -\frac{2}{3} \) and y-intercept \( b = -3 \).\For the second equation \( y - \frac{1}{3}x = 6 \), add \( \frac{1}{3}x \) to both sides to get: \\[ y = \frac{1}{3}x + 6 \] \This equation is in slope-intercept form \( y = mx + b \), with slope \( m = \frac{1}{3} \) and y-intercept \( b = 6 \).

Step 2: Plot the graphs of the equations

We will plot both equations on a coordinate plane.\For the first equation \( y = -\frac{2}{3}x - 3 \): \- Start at the y-intercept \( (0, -3) \). \- Use the slope \( -\frac{2}{3} \) to plot another point. From the y-intercept, move down 2 units and right 3 units to plot another point.\For the second equation \( y = \frac{1}{3}x + 6 \):\- Start at the y-intercept \( (0, 6) \). \- Use the slope \( \frac{1}{3} \) to plot another point. From the y-intercept, move up 1 unit and right 3 units to plot another point.\Draw straight lines through the plotted points for each equation.

Step 3: Identify the point of intersection

Observe where the two lines intersect on the graph. The coordinates of the intersection point represent the solution to the system of equations.\In this case, the lines representing the equations \( y = -\frac{2}{3}x - 3 \) and \( y = \frac{1}{3}x + 6 \) intersect at the point \((-9, 3)\). Thus, the solution to the system is \( x = -9 \) and \( y = 3 \).

Key Concepts

Graphing Linear Equations

Graphing linear equations involves plotting one or more lines on a coordinate plane. Each line represents a solution set of a linear equation. To create the graph, we often convert the equation to its slope-intercept form. This equation format tells us how to plot the graph quickly and efficiently.

• Start by identifying the y-intercept, which is the point where the line crosses the y-axis. This is always at \((0, b)\), with \(b\) being the y-intercept value.
• Next, use the slope of the line to determine additional points. The slope \((m)\) is a crucial piece; it indicates how steep the line is, and describes the change in \(y\) per unit change in \(x\). A positive slope means the line ascends, whereas a negative slope means it descends.
• After plotting these points, draw a straight line through them.

Graphing is especially helpful for visualizing solutions. When you have a system of equations, each one gets its line on the graph, and where these lines intersect is key.

Slope-Intercept Form

The slope-intercept form of a linear equation is a very user-friendly format. It's written as \( y = mx + b \), where \(m\) is the slope of the line and \(b\) is the y-intercept. This form is most useful for graphing and provides clear information:

• **Slope \(m\):** It tells us the rate of increase or decrease. A larger absolute value of \(m\) means a steeper slope.
• **Y-Intercept \(b\):** Marks where the line crosses the y-axis. From here, you use the slope to find other points on the graph.

For the equations in our exercise, converting them to slope-intercept form simplifies everything. You simply find the slope and the intercept, plot the y-intercept, and use the slope to find another point. Remember, understanding these component parts makes graphing much more intuitive.

Point of Intersection

In the graph of a system of linear equations, the point of intersection is a pivotal point. It represents the solution that satisfies all equations in the system simultaneously.

• **Finding the Intersection:** On a graph, this is where the lines cross one another. The coordinates of this point correspond to the values that make both equations true.
• **Significance:** Knowing this point lets us solve systems of equations visually, confirming algebraic solutions. In our exercise, the lines of the two equations intersect at \((-9, 3)\).
• **Practical Uses:** Points of intersection are used extensively in various fields—engineering, economics, and even in simple everyday problems—to find where conditions align.

Remember, when you graph the system, the clearer your graph, the easier it is to identify the point of intersection and hence the solution of the system.