Problem 68 - Elementary Algebra Within Reach

Question

In Exercises \(67-70\), solve the system by graphing. $$ \left\\{\begin{array}{l} x-2 y=6 \\ x+2 y=2 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x=4, y=-1\)

Step 1: Rewrite the Equations in Slope-Intercept Form

To plot a line, it's often useful to write the equation in slope-intercept form \(y = mx + b\), where m is the slope of the line and b is the y-intercept. So, for the given equations: (1) \(x - 2y = 6\) can be written as \(y = 0.5*x - 3\) (2) \(x + 2y = 2\) can be written as \(y = -0.5*x + 1\)

Step 2: Graph the Lines

Plot the y-intercepts first. For line (1), b equals -3, so the y-intercept is at point (0,-3). For line (2), b equals 1, so the y-intercept is at point (0,1). Then, use the slope to identify another point on each line and draw the lines.

Step 3: Identify the Intersection Point

The solution to the system is at the point where the two lines intersect. Compare the two lines and it is evident that they intersect at the point \(x=4, y=-1\). This is the solution to the system.

Key Concepts

Graphing Linear Equations

Graphing linear equations is an easy and intuitive method to visualize how different equations relate to one another. When you graph equations, you're actually plotting a series of coordinates on a Cartesian plane.
This serves as a crucial step in solving a system of linear equations.

• First, you should rewrite each equation in a manageable form, typically as a linear equation.
• Then, translate these algebraic equations into visual lines on a graph.
• The simplicity of graphing lies in the use of graph paper or a digital tool, which makes plotting precise.

The main goal when graphing is to find out where the lines cross or intersect, and this point is what's called the solution to the equation system.

Slope-Intercept Form

The slope-intercept form is a common way to express linear equations, written as \(y = mx + b\).
Here, \(m\) represents the slope, and \(b\) is the y-intercept, or where the line crosses the y-axis.
This form is very handy when graphing, because it provides an immediate understanding of the line's direction and starting point.

• The slope \(m\) indicates how steep the line is and the direction (upward or downward) across the graph.
• The y-intercept \(b\) helps you start plotting the line with a clear point on the y-axis.

For instance, in our problem, two equations were first rewritten in this form to prepare for graphing. The ability to rearrange equations into slope-intercept form is key to drawing accurate graphs.

Intersection of Lines

The intersection of lines on a graph reveals the solution to a system of linear equations. When two lines cross, the coordinates of the intersection represent values of \(x\) and \(y\) that satisfy both equations simultaneously.
Finding this point is the ultimate goal when using graphing to solve systems.

• Identify the point where both lines share the same coordinate values.
• This intersection represents a valid solution to both linear equations.
• Sometimes, the intersection may be a decimal or fraction, requiring careful graph examination.

In the assignment exercise, identifying the precise intersection point \((4, -1)\) was the solution to the system. Being able to spot and verify this point is crucial for accurately solving any system using the graphing method.