Problem 35 - Algebra A Combined Function

Question

Solve each system of linear equations by graphing. See Examples 3 through \(6 .\) \(\left\\{\begin{array}{l}2 x-3 y=-2 \\ -3 x+5 y=5\end{array}\right.\)

Step-by-Step Solution

Answer

The lines intersect at the point \((5, 4)\), which is the solution.

Step 1: Rewrite Each Equation in Slope-Intercept Form

First, take the equation \(2x - 3y = -2\) and solve for \(y\):\[3y = 2x + 2\] \[y = \frac{2}{3}x + \frac{2}{3}\]. Next, take the second equation, \(-3x + 5y = 5\) and solve for \(y\): \[5y = 3x + 5\] \[y = \frac{3}{5}x + 1\].

Step 2: Graph Each Equation on the Same Coordinate Plane

Graph the first equation, \(y = \frac{2}{3}x + \frac{2}{3}\), by plotting the y-intercept \((0, \frac{2}{3})\) and using the slope \(\frac{2}{3}\) to find another point, moving up 2 units and right 3 units. Similarly, graph the second equation, \(y = \frac{3}{5}x + 1\), by plotting the y-intercept \((0, 1)\) and using the slope \(\frac{3}{5}\) to find another point, moving up 3 units and right 5 units.

Step 3: Determine the Point of Intersection

Examine where the two lines intersect on the graph. The intersection represents the solution to the system of equations. Upon graphing, the two lines intersect at the point \((5, 4)\).

Step 4: Verify the Solution by Substitution

Substitute the intersection point \((5, 4)\) back into the original equations to ensure they hold true. For \(2x - 3y = -2\): \(2(5) - 3(4) = 10 - 12 = -2\), which is correct. For \(-3x + 5y = 5\): \(-3(5) + 5(4) = -15 + 20 = 5\), which is also correct.

Key Concepts

Graphing

Graphing is a visual method for solving systems of linear equations. It involves drawing each equation as a line on a coordinate grid to find the solution where they intersect. To graph effectively, it's crucial to understand your grid and axes:

• Each line represents one equation from the system.
• Plot each line carefully, using the y-intercept and slope for guidance.
• Mark the intersection point clearly where the lines meet. This spot is your solution.

Understanding how to graph correctly helps you visualize complex problems simply. When done accurately, it provides a clear picture of the relationship between variables, representing all solutions to the equation.

Intersection Point

The intersection point of two lines on a graph is where the lines cross. This point has key significance in a system of linear equations. It represents the values of the variables that solve both equations simultaneously. To identify the intersection point:

• Ensure both lines are accurately plotted.
• Pinpoint where the two lines meet.
• The coordinates at this spot are your solutions for the system.

Checking the intersection point by substitution into both original equations validates the solution. In our example, the intersection point is \(5, 4\), where both equations are true when these values are plugged in.

Slope-Intercept Form

The slope-intercept form is a way of writing the equation of a line so that it can be easily graphed. This form is written as \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept:

• The slope \(m\) shows the tilt of the line and is calculated as rise over run.
• The y-intercept \(b\) gives the point where the line crosses the y-axis.

Converting equations to this form simplifies the process of graphing lines on a coordinate plane. It allows you to quickly identify the starting point and how to move to plot the line correctly. In our exercise, both equations were converted to slope-intercept form to facilitate easy graphing and thus solving.