Problem 35 - Elementary Algebra Within Reach

Question

In Exercises 27-36, solve the system by graphing. $$ \left\\{\begin{array}{r} 8 x-6 y=-12 \\ x-\frac{3}{4} y=-2 \end{array}\right. $$

Step-by-Step Solution

Answer

The system of equations has no solution.

Step 1: SetUp Equations

To rearrange the equations into the slope-intercept form (y = mx + c) we need to isolate 'y' in each equation. The given equations are: \(8x - 6y = -12\) and \(x - \frac{3}{4}y = -2\).

Step 2: Rearrange Equation 1

Rearrange \(8x - 6y = -12\) to isolate 'y'. Add '6y' on both sides and subtract '-12' on both sides which leads to \(6y = 8x + 12\). Now divide both sides by 6, which gives \(y = \frac{4}{3}x + 2\). Equation for First line is hence \(y = \frac{4}{3}x + 2\).

Step 3: Rearrange Equation 2

Rearrange \(x - \frac{3}{4}y = -2\) to isolate 'y', it becomes \(\frac{3}{4}y = x + 2\). Multiply all terms by 4/3 to get rid of fractions which gives \(y = \frac{4}{3}x + \frac{8}{3}\). Equation for Second line is \(y = \frac{4}{3}x + \frac{8}{3}\).

Step 4: Graphing

Draw the graphs of both lines on the same Cartesian plane. Because they possess same slope but different y-intercepts, they are parallel to each other, and therefore, won't intersect.

Step 5: Conclusion

Since the lines do not intersect, there is no solution for the system. This means the two equations are inconsistent and no common solution that satisfies both equations at the same time exists.

Key Concepts

Slope-Intercept Form

When solving linear systems by graphing, converting the equations into the slope-intercept form, denoted as \(y = mx + c\), can be extremely helpful. This format allows us to easily identify the slope of the line \(m\) and the y-intercept \(c\).

• Slope (\(m\)): It represents the steepness or inclination of the line. A positive slope means the line rises, while a negative slope means it falls as it moves from left to right.
• Y-intercept (\(c\)): This is the point where the line crosses the y-axis. It's the value of \(y\) when \(x\) is zero.

In our exercise, the two equations were rearranged into slope-intercept form to better understand their characteristics:

• First line: \(y = \frac{4}{3}x + 2\)
• Second line: \(y = \frac{4}{3}x + \frac{8}{3}\)

Both lines have a slope of \(\frac{4}{3}\), which is key to discovering their relationship.

Parallel Lines

Parallel lines exhibit a specific characteristic: they never meet. This means they will never intersect, no matter how far they are extended in either direction. When graphing linear equations, a key indicator that lines are parallel is if they share the same slope (\(m\)), as seen in the slope-intercept form.
In the exercise provided, both equations shared the same slope of \(\frac{4}{3}\). However, the y-intercepts were different:

• First line y-intercept: 2
• Second line y-intercept: \(\frac{8}{3}\)

The equivalent slopes but differing y-intercepts illustrate why these lines are parallel and hence do not share any points of intersection on the graph.

Inconsistent Equations

In systems of equations, consistency refers to the existence of a solution that satisfies all equations in the system. An inconsistent system, on the other hand, means that no solution can satisfy all the equations simultaneously.

• Graphical Interpretation: If two lines are parallel, they never intersect, and thus no shared solution exists.
• Algebraic Indicator: In slope-intercept form, lines having identical slopes but different y-intercepts confirm inconsistency.

In our exercise, the given lines were parallel due to identical slopes, \(\frac{4}{3}\), yet different y-intercepts, which dictated that the system of equations is inconsistent. Therefore, no point exists that would satisfy both equations at the same time. This means there is no solution to the system of equations.