Problem 60 - A Graphical Approach to College Algebra

Question

Solve each system graphically. Check your solutions. Do not use a calculator. $$\begin{array}{r}3 x-y=4 \\\x+y=0\end{array}$$

Step-by-Step Solution

Answer

The solution is (1, -1).

Step 1: Rewrite Equations in Slope-Intercept Form

To graph the system of equations, we first need to rewrite each equation in the slope-intercept form, which is y = mx + b. For the first equation, 3x - y = 4: 1. Move 3x to the other side: -y = -3x + 4. 2. Multiply through by -1: y = 3x - 4. For the second equation, x + y = 0: 1. Subtract x from both sides: y = -x.

Step 2: Graph the First Equation

The first equation is y = 3x - 4. This line has a slope of 3 and a y-intercept of -4. 1. Plot the y-intercept (0, -4) on the graph. 2. Using the slope of 3, move up 3 units and right 1 unit to plot the next point. 3. Draw a line through these points extending in both directions.

Step 3: Graph the Second Equation

The second equation is y = -x. This line has a slope of -1 and a y-intercept of 0. 1. Plot the y-intercept (0, 0) on the graph. 2. Using the slope of -1, move down 1 unit and right 1 unit to plot the next point. 3. Draw a line through these points extending in both directions.

Step 4: Identify the Intersection Point

Look where the two lines intersect on the graph. They meet at the point (1, -1). This point is the solution to the system of equations because it satisfies both equations.

Step 5: Check the Solution Algebraically

Substitute (1, -1) into both original equations to confirm it is a solution: 1. For 3x - y = 4: Substitute x=1 and y=-1: 3(1) - (-1) = 3 + 1 = 4, which is correct. 2. For x + y = 0: Substitute x=1 and y=-1: 1 + (-1) = 0, which is also correct. Both equations hold true, confirming (1, -1) is the solution.

Key Concepts

Slope-intercept form

The slope-intercept form is a method to express the equation of a straight line. It is written as \( y = mx + b \) where:

• \(m\) is the slope of the line.
• \(b\) is the y-intercept, which is the point where the line crosses the y-axis.

This form allows you to quickly identify the slope and intercept, making it easier to graph the line.
In the problem, both equations were transformed into slope-intercept form:

• From \(3x - y = 4\) to \(y = 3x - 4\)
• From \(x + y = 0\) to \(y = -x\)

The slope in the first equation, \(3x - y = 4\), is 3, meaning the line rises 3 units for every 1 unit you move to the right. For the second equation, \(x + y = 0\), the slope is -1, indicating the line falls 1 unit for every 1 unit you move to the right.
This form is a powerful tool in graphing because it simplifies the understanding of the line's direction and position.

System of equations

A system of equations is a collection of two or more equations with the same set of variables. In this exercise, we are looking at a system with two equations and two unknowns (\(x\) and \(y\)):

• \(3x - y = 4\)
• \(x + y = 0\)

Solving a system of equations means finding the values of the variables that satisfy all equations simultaneously. There are several methods to solve a system of equations:

• Graphically
• Substitution
• Elimination

In this exercise, we are focusing on solving it graphically. The solution to the system is the point on the graph where both lines intersect.
Here, the intersection point is \((1, -1)\), which means \(x = 1\) and \(y = -1\) satisfy both equations.
Checking your solution by substituting these values back into the original equations verifies the result.

Graphing linear equations

Graphing linear equations involves drawing the line that represents each equation on a coordinate plane. This is done by:

• Identifying the y-intercept.
• Using the slope to find additional points.
• Drawing a line through these points.

For the first equation, \(y = 3x - 4\), the y-intercept is -4. Start by plotting this point, \((0, -4)\). From there, use the slope of 3 to find another point by moving up 3 units and right 1 unit, then draw the line through these points extending in both directions.
The second equation, \(y = -x\), has a y-intercept of 0. Plot \((0,0)\), then use the slope of -1: move down 1 unit and right 1 unit to locate the second point.
Drawing lines for these equations visually shows where they cross. The intersection of these lines is the solution to the system of equations. In this case, the point \((1, -1)\) is where the two lines meet, confirming it as the solution.