Problem 34 - Precalculus with Limits

Question

Solving a System of Equations Graphically In Exercises \(33-44,\) solve the system graphically. $$\left\\{\begin{array}{c}{x+y=0} \\ {3 x-2 y=5}\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is the coordinates of the intersection point of the two graphs. Verify this by substituting the x and y values into both of the original equations to make sure it satisfies both.

Step 1: Plot the First Equation

This equation is: \(x + y = 0\) which can be rewritten as \(y = -x\). You can start by plotting the line \(y = -x\). Use the y-intercept and slope to guide you. The y-intercept is 0 (the point where the line crosses the y-axis), and the slope is -1, meaning that for each positive step along the x axis, we step one unit down on the y axis.

Step 2: Plot the Second Equation

The second equation is: \(3x - 2y = 5\) which can be rewritten as \(y = \frac{3}{2}x - \frac{5}{2}\). Start by marking the y-intercept (-5/2) on the y-axis. The slope is 3/2, meaning that for every two units you move to the right on the x-axis, move three units up on the y-axis.

Step 3: Identify the Intersection Point

The intersection point of these two lines represents the solution for the system of equations. Locate this point and identify its coordinates as \((x, y)\).

Step 4: Verify the solution

Substitute the x and y values from the intersection point into both the original equations to ensure it satisfies both, confirming it as the solution to the system of equations.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that create straight lines when graphed on a coordinate plane. They follow the form \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants. The equation describes a relation where the change in one variable is proportional to the change in another.
In our exercise, we have two linear equations: \( x + y = 0 \) and \( 3x - 2y = 5 \). The first one can be rewritten as \( y = -x \), a familiar form for graphing, while the second can be rewritten as \( y = \frac{3}{2}x - \frac{5}{2} \).
Understanding each equation's format simplifies plotting them on a graph and finding solutions where they intersect. This graphical approach helps visualize how different relationships between variables can coexist.

Intersection Point

When working with systems of equations, the intersection point is a crucial concept. This point represents the single set of values for \( x \) and \( y \) that satisfy both equations. Graphically, it is where two lines cross each other on a coordinate plane.
In our exercise, after graphing the two lines, the intersection point can be observed at their crossing. Finding it involves visually identifying the coordinates at which both lines meet. This point is a solution that meets the conditions defined by both equations simultaneously.
To ensure accuracy, it's good practice to solve for the intersection point mathematically after identifying it graphically. Verify by substituting the \( x \) and \( y \) values back into the original equations to check if both are satisfied.

Slope and Intercept

The slope and intercept of a line reveal much about its direction and position on a graph. The slope describes the steepness and direction of a line, defined as the ratio of the change in \( y \) over the change in \( x \). Meanwhile, the intercepts show where the line crosses the axes.
In our exercise, the first equation \( y = -x \) has a slope of \(-1\), indicating a downward slope where every one unit increase in \( x \) necessitates a one unit decrease in \( y \). The y-intercept is 0, so the line passes through the origin.
The second equation \( y = \frac{3}{2}x - \frac{5}{2} \) has a slope of \( \frac{3}{2} \), suggesting a relatively steep upward direction. Its y-intercept is \(-\frac{5}{2} \), meaning it crosses the y-axis below the origin.

• To plot these equations, start with the intercept, then use the slope to determine additional points along the line.
• Understanding slope and intercept is key to graphing and solving systems of equations efficiently.