Problem 39 - College Algebra and Calculus: An Applied Approach

Question

Solve the system graphically. $$\left\\{\begin{array}{r}-x+2 y=2 \\ 3 x+y=15\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system is the point of intersection which, from the graphical solution, is found to be (3, 6).

Step 1: Transform Equations To Slope-Intercept Form

First, both equations must be written in slope-intercept form (y = mx + c). \nFor the first equation: \n- Arrange -x + 2y = 2 to obtain y = 1/2x + 1. \nFor the second equation: \n- Arrange 3x + y = 15 to get y = -3x + 15.

Step 2: Plotting The Equations

Next, draw the graphs for both equations on the same plane using their slope-intercept forms. The line for the first equation, y = 1/2x + 1, starts from y-intercept (1) and is positively sloped. The line for the second equation, y = -3x + 15, starts from y-intercept(15) and is negatively sloped.

Step 3: Find The Intersection Point

Where both lines intersect is the solution of the system as it is the value that satisfies both equations. With the help of graph, it appears intersection point is at (3,6). However, always check this solution by substituting x as 3 in both equations to ensure that in both cases, y = 6.

Key Concepts

Understanding Slope-Intercept Form

One of the most efficient ways to represent a linear equation is using the slope-intercept form, which is expressed as
\[ y = mx + b \],
where \( m \) is the slope of the line and \( b \) is the y-intercept. The slope quantifies how steep the line is, and the y-intercept specifies where the line crosses the y-axis on a graph.

Breaking Down the Slope

The slope, represented by \( m \), indicates the rise over run, meaning how much the line goes up (or down) for a certain horizontal distance along the x-axis. A positive slope means the line ascends from left to right, while a negative slope descends.

Identifying the Y-Intercept

The y-intercept \( b \) is simply the point where the line meets the y-axis. This is the value of \( y \) when \( x = 0 \). Understanding these concepts is essential for graphing linear equations and helps immensely in visualizing and solving systems of equations.

Graphing Linear Equations

Graphing is a powerful tool that uses visual representation to solve equations. To graph a linear equation, follow these steps:

• First, write the equation in slope-intercept form \( y = mx + b \).
• Identify the slope \( m \) and y-intercept \( b \) from the equation.
• Plot the y-intercept on the graph as a starting point.
• Use the slope to determine the direction and angle of the line by moving from the y-intercept up/down and right.
• Draw a straight line through these points, extending it across the grid.

With practice, you'll get faster at sketching these lines and visualizing the graph of the corresponding equation.

Consistent Practice

Often, students find it helpful to practice with different slopes and intercepts to build intuition and speed in graphing linear equations—a crucial step in finding solutions for systems of equations graphically.

Finding the Intersection Point of Lines

The intersection point of two lines on a graph is significant because it represents a pair of x and y values that satisfy both equations in a system. Here's how you find it:

• Graph both equations on the same coordinate plane using the steps for graphing linear equations.
• Look for the point where the two lines cross each other—it's the intersection point.
• Read the coordinates of the intersection point directly from the graph.
• Always verify by substituting the coordinates into both original equations to ensure they satisfy both.

Verification Is Key

Verification ensures that the graphical representation is precise, as graphing by hand can sometimes lead to small inaccuracies. In the given exercise, after graphing both equations and finding the intersection at (3,6), substituting those values back into the original equations must confirm that point as the solution. This step is crucial for ensuring accuracy in your graphical solutions.