Problem 30 - Intermediate Algebra

Question

Solve each system by graphing. See Example 5 . $$ \left\\{\begin{array}{l} x=y+3 \\ \frac{1}{4} x-\frac{1}{6} y=\frac{1}{3} \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is the point \( (6, 3) \).

Step 1: Understand the System of Equations

We need to solve the system of equations by graphing. The system is: \( x = y + 3 \) and \( \frac{1}{4}x - \frac{1}{6}y = \frac{1}{3} \).

Step 2: Rearrange First Equation

The first equation is already solved for \( x \), which gives us: \( x = y + 3 \). We can rewrite it as \( y = x - 3 \) for easier graphing.

Step 3: Rearrange Second Equation for y

Start with the second equation: \( \frac{1}{4}x - \frac{1}{6}y = \frac{1}{3} \).- Multiply every term by 12 to clear the fractions: \( 3x - 2y = 4 \).- Solve for \( y \): - Add \( 2y \) to both sides: \( 3x = 2y + 4 \). - Subtract 4: \( 3x - 4 = 2y \). - Divide by 2: \( y = \frac{3}{2}x - 2 \).

Step 4: Plot the Equations on a Graph

Graph both equations on the same coordinate plane:- For \( y = x - 3 \): This is a line with a slope of 1 and y-intercept at -3.- For \( y = \frac{3}{2}x - 2 \): This is a line with a slope of \( \frac{3}{2} \) and y-intercept at -2.

Step 5: Find the Intersection Point

The solution of the system is the intersection point of the two lines. - Plotting both lines, observe where they intersect. The lines intersect at the point (6, 3).

Step 6: Verify the Solution

Verify by plugging the point (6, 3) into both original equations:- For \( x = y + 3 \), substitute getting \( 6 = 3 + 3 \), which is true.- For \( \frac{1}{4}x - \frac{1}{6}y = \frac{1}{3} \), substitute getting \( \frac{1}{4}(6) - \frac{1}{6}(3) = \frac{1}{3} \), simplifying to \( \frac{3}{2} - \frac{1}{2} = 1 \), which equals \( \frac{1}{3} \).

Key Concepts

Graphing Linear Equations

Graphing linear equations is a visual method used to solve systems of equations. Each equation in the system represents a straight line on the graph.
By plotting these lines, we can easily observe where they intersect, which represents the solution to the system. The first step is to convert each equation into the form of \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
The slope indicates how steep the line is and the direction it goes, while the y-intercept is the point where the line crosses the y-axis. Understanding these concepts is crucial for accurately graphing the lines and finding their intersection point.

Intersection Point

The intersection point is where two or more graphs meet on the coordinate plane. In the context of a system of equations, this point is the solution that satisfies both equations simultaneously.
For linear equations, the intersection point can often be found by graphing both lines and observing their meeting point.
In the exercise, the intersection point was found to be \((6, 3)\). This point, when substituted back into the original equations, should satisfy both completely. If it doesn't, one or more errors in graphing or calculation might have occurred.

Solving by Graphing

Solving systems of equations by graphing is a straightforward method, especially when dealing with linear equations. Start by graphing each line on the same set of axes.
To graph, utilize each equation's slope and y-intercept, placing points accordingly. Then draw a straight line through these points.
Once both lines are drawn, identify the intersection point. This solution should be verified by substituting it back into the original equations to confirm that it satisfies both.
While solving by graphing is visually intuitive, it's important to note that precise graphing is crucial to accurately determine the intersection point, especially when dealing with fractional coefficients.

Rearranging Equations

Rearranging equations is an essential skill when solving systems of equations by graphing. Often, equations are not initially in a simple form that’s easy to graph.
Rearranging aims to express the equation in the form \( y = mx + b \). In the original exercise, the first equation, \( x = y + 3 \), was rearranged to \( y = x - 3 \) to make the form clear for graphing.
The second equation required clearing fractions by multiplying all terms by a common denominator, leading finally to \( y = \frac{3}{2}x - 2 \).
This preparation step simplifies graphing by clearly identifying the slope and y-intercept needed to draw the straight line on a graph.