Problem 40 - College Algebra and Calculus: An Applied Approach

Question

Solve the system graphically. $$\left\\{\begin{aligned} x+y &=0 \\ 3 x-2 y &=10 \end{aligned}\right.$$

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: Convert the equations to slope-intercept form

The given system of equations is: \[\begin{aligned} x+y &=0 \\ 3x-2y &=10 \end{aligned}\] To convert each equation to slope-intercept form, solve for \(y\). So, the new system of equations becomes: \[\begin{aligned} y &= -x \\ y &= \frac{3x}{2} - 5 \end{aligned}\]

Step 2: Graph the lines

Now, graph both these lines on a coordinate plane. The first line depicts a straight line passing through the origin (0,0) with a slope of -1. The second line will cut the y-axis at the point (0,-5) and the graph will be a straight line rising from left to right with a positive slope of 1.5.

Step 3: Identify the intersection point

Locate the point where the two lines intersect. Each line represents the solutions of their respective equations — so the point where they intersect represents the solution of both equations, thereby solving the system.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most common and practical forms used in algebra.
It is expressed as \( y = mx + b \), where:

• \( m \) represents the slope of the line, indicating its steepness and direction.
• \( b \) represents the y-intercept, the point where the line crosses the y-axis.

To convert any linear equation into slope-intercept form, you need to rearrange it to solve for \( y \).
In doing so, you can easily visualize the line's behavior on a graph. For instance, in our example, converting \( x + y = 0 \) gives us \( y = -x \), showing the line with a slope of -1 and a y-intercept at the origin (0,0).
Similarly, transforming the second equation \( 3x - 2y = 10 \) into \( y = \frac{3}{2}x - 5 \) offers a clear slope of 1.5, with a y-intercept at (0,-5).
By mastering this conversion, it becomes straightforward to graph the lines and comprehend their relationships.

Intersection Point

The intersection point is fundamental when solving a system of equations graphically.
It is the point where two lines on a graph cross each other, representing the solution to both equations in the system.
In the example provided, after plotting both equations on the coordinate plane, we aim to find where the lines \( y = -x \) and \( y = \frac{3}{2}x - 5 \) intersect.
This intersection is essentially the values of \( x \) and \( y \) that satisfy both equations simultaneously.Finding the intersection typically involves looking for the common solution, or shared coordinate pair \((x, y)\), of the two lines.
Once you identify this spot on the graph, you have graphically solved the system of equations.
The intersection point provides a tangible solution that can be checked by substituting back into the original equations to verify its validity.

Coordinate Plane

The coordinate plane is an essential tool in mathematics for graphically solving equations.
It consists of two perpendicular axes: the horizontal x-axis and the vertical y-axis, intersecting at the origin point, (0,0).
Every point on the plane can be identified by a pair of numbers (x, y), known as coordinates, indicating the point's location relative to the axes. When graphing equations, the coordinate plane allows us to visualize how different lines, curves, or shapes sit in relation to each other.
For linear equations in slope-intercept form, the graph appears as a straight line across the plane.
In our example, plotting the equations onto this plane reveals the lines we discussed:

• The first line crosses the origin with a negative slope, creating a downward slant.
• The second line intersects the y-axis at -5 and slants upward with positive slope.

By using the coordinate plane effectively, you can see these interactions and find solutions such as the intersection point much more clearly.