Problem 33 - Precalculus with Limits

Question

Solving a System of Equations Graphically In Exercises \(33-44,\) solve the system graphically. $$\left\\{\begin{array}{c}{-x+2 y=-2} \\ {3 x+y=20}\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the given system of equations is the point (4, 2).

Step 1: Rearrange each equation in y = mx + c format

First, rearrange the two equations to have them in slope-intercept form or y = mx + c format. This format makes it easier to plot the graph of each line. \[ y = \frac{1}{2}x + 1 \] and \[ y = 20 - 3x \]

Step 2: Graph each equation

Second step is graphing each equation on the same set of axes. The first line has a slope of 0.5 (which means it rises 1 and runs 2) and a y-intercept of 1. The second line has a slope of -3 and y-intercept of 20, it goes down steeply from the point (0,20) on the Y-axis. Plot both these lines on the graph.

Step 3: Identify the point of intersection

The point where two lines intersect each other is the solution to the system of equations. The lines will intersect at point (4,2).

Key Concepts

Understanding Slope-Intercept Form

When solving systems of equations, particularly graphing linear equations, it's beneficial to first convert the equations into slope-intercept form, which is given by the equation \( y = mx + c \). Here, \(m\) represents the slope of the line, and \(c\) denotes the y-intercept, which is where the line crosses the y-axis.

The slope \(m\) tells us how steep the line is. A positive slope means the line rises as it moves from left to right, while a negative slope indicates it falls. The numerical value of \(m\) shows how much the line goes up (or down) for every unit it goes to the right. For example, a slope of 2 means the line rises 2 units for every 1 unit it moves horizontally.

The y-intercept \(c\) is straightforward as well: it's the point where the line crosses the y-axis. This value tells us where the graph of the equation begins (or ends) vertically on the graph. By knowing \(m\) and \(c\), we can draw the entire line on the graph, starting from the y-intercept and following the slope. If you struggle with numbers, remember that the higher \(c\) is, the further up the y-axis our line starts.

Graphing Linear Equations Effectively

To graph a linear equation, start by plotting the y-intercept on the graph. This is your starting point. Now, use the slope to determine the direction of the line. If the slope is written as a fraction, such as \(\frac{1}{2}\), the numerator tells you how many units to go up (positive slope) or down (negative slope), and the denominator tells you how many units to go to the right.

For instance, let's plot the equation \(y = \frac{1}{2}x + 1\). We start at the y-intercept (0,1) and use the slope of \(\frac{1}{2}\) to rise 1 unit and run 2 units to plot the next point on the graph. For a line like \(y = 20 - 3x\), we'd start at (0,20) and because the slope is \( -3 \), we'd go down 3 units for every 1 unit we move to the right.

Tips for Accurate Graphing

• Always start with plotting the y-intercept.
• Use the slope to find another point on the line.
• Draw a straight line through the points, extending it in both directions.
• Check your work by ensuring that the slope between any two points on the line matches the slope given in the equation.

Remember, accuracy is key in graphing, so take your time to carefully plot the points and draw the lines.

Identifying the Point of Intersection

The point of intersection is where the graphs of two linear equations meet on the coordinate plane. This point is significant because it represents the solution to the system of equations, meaning these are the x and y values that satisfy both equations simultaneously.

To find the point of intersection graphically, draw the lines represented by each equation on the same graph. Look for the point where both lines cross. The coordinates of this point are your solution. For example, if the lines intersect at the point \( (4,2) \), then \( x = 4 \) and \( y = 2 \) are the solutions to the system of equations.

How to Ensure Accuracy

• Make sure both lines are graphed accurately.
• Double-check the slope and y-intercept for each line.
• If the lines don't appear to cross, re-evaluate your graph as it's likely a mistake has been made.
• In some cases, lines may be parallel (no intersection) or coincident (infinite intersections), indicating different types of solutions or no solution.

Understanding the point of intersection can help you verify the solutions to systems of linear equations not just graphically, but when using other methods like substitution or elimination as well.