Problem 35 - Algebra 2

Question

Solve each system of equations by graphing. \(\frac{4}{3} x+\frac{1}{5} y=3\) \(\frac{2}{3} x-\frac{3}{5} y=5\)

Step-by-Step Solution

Answer

The solution is the point where the two lines intersect on the graph.

Step 1: Convert Equations to Slope-Intercept Form

To graph the equations, it's helpful to convert them to the slope-intercept form \( y = mx + b \). For the first equation, \( \frac{4}{3} x + \frac{1}{5} y = 3 \), move \( \frac{4}{3} x \) to the other side: \( \frac{1}{5} y = -\frac{4}{3} x + 3 \). Multiply everything by 5 to clear fractions: \( y = -\frac{20}{3} x + 15 \). For the second equation, \( \frac{2}{3} x - \frac{3}{5} y = 5 \), move \( \frac{2}{3} x \) to the other side: \( -\frac{3}{5} y = -\frac{2}{3} x + 5 \). Multiply each term by \(-\frac{5}{3}\) to solve for \( y \): \( y = \frac{10}{9} x - \frac{25}{3} \).

Step 2: Graph the Equations

Graph both equations by plotting the y-intercept and using the slope to find other points. For the first equation \( y = -\frac{20}{3} x + 15 \), the y-intercept is 15, and the slope is -\(\frac{20}{3}\). For every 3 units moved to the right, go down 20 units. For the second equation \( y = \frac{10}{9} x - \frac{25}{3} \), the y-intercept is \(-\frac{25}{3}\) which is approximately \(-8.33\), and the slope is \(\frac{10}{9}\). For every 9 units moved to the right, go up 10 units.

Step 3: Identify the Intersection Point

The solution to the system is the point where the two lines intersect on the graph. From the graph, you can see the intersection point is the solution to the system of equations.

Key Concepts

Graphing Linear Equations

When you graph a linear equation, you are drawing a straight line on a grid. This grid usually has two axes: the horizontal axis, known as the x-axis, and the vertical axis, called the y-axis. The purpose of graphing linear equations is to visually represent the solutions to the equation.

To start graphing, you need at least two points that satisfy the equation. Typically, you find these points by determining the y-intercept and using the slope to find another point.

• Draw the x and y axes on a piece of graph paper.
• Plot the y-intercept, which is where the line crosses the y-axis.
• Use the slope of the equation to locate additional points and draw the line.

Once you have multiple points, connect them with a straight line to complete the graph. This method is used for each equation in the system to visualize how they interact with one another.

Slope-Intercept Form

The slope-intercept form of a linear equation is expressed as \( y = mx + b \), where \( m \) is the slope of the line, and \( b \) is the y-intercept.

This form is particularly valuable because it provides clear information about the line's direction and where it crosses the y-axis.

• Slope \( m \): Indicates the steepness and direction of the line. A positive slope means the line goes upwards as it moves to the right, while a negative slope means it descends.
• Y-Intercept \( b \): Shows the point at which the line crosses the y-axis. This point has an x-coordinate of 0.

To convert from standard form like \( Ax + By = C \), isolate \( y \) on one side. For example, in the equation \( \frac{4}{3}x+\frac{1}{5}y=3 \), by rearranging terms and solving for \( y \), you convert it into the slope-intercept form.

Intersection Point in Graphing

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

Here’s why identifying the intersection point is important:

• The coordinates of this point satisfy both equations in the system.
• It visually illustrates where both conditions or relationships (represented by each line) hold true simultaneously.

To find this point, graph each equation on the same set of axes and closely observe where they meet. This meeting point can usually be read directly from the graph. However, if the lines happen to be parallel, they do not intersect, indicating there is no solution. On the other hand, if they represent the same line, their intersection is infinite, meaning the system has many solutions.