Problem 48 - Elementary Algebra Within Reach

Question

In Exercises \(43-48\), write the equations of the lines in slope-intercept form. What can you conclude about the number of solutions of the system? $$ \left\\{\begin{array}{r} 7 x+6 y=-4 \\ 3.5 x+3 y=-2 \end{array}\right. $$

Step-by-Step Solution

Answer

The lines represented by the equations are parallel, thus the system has no solutions.

Step 1: Convert the first equation into slope-intercept form

The given equation is 7x+6y=-4. Isolate the term 'y'. Divide every term in the equation by 6 to get y by itself: y = -7/6 * x - 4/6

Step 2: Convert the second equation into slope-intercept form

The second equation is 3.5x+3y=-2. Isolate the term 'y'. Divide every term in the equation by 3 to get y by itself: y = -7/6 * x - 2/3

Step 3: Comparing the two equations

We can observe that both equations have the same slope (-7/6). However, their y-intercepts are different (-4/6=-2/3 and -2/3, respectively). This means the lines are parallel.

Step 4: Conclusion

Parallel lines do not intersect, which means there are no common solutions to these equations. Therefore, it can be concluded that the system of equations has no solutions.

Key Concepts

System of Equations

A system of equations consists of a set of two or more equations that share common variables. The goal is to find values for these variables that satisfy all the equations simultaneously. In our example, we have:

• Equation 1: \( 7x + 6y = -4 \)
• Equation 2: \( 3.5x + 3y = -2 \)

Each equation represents a line when plotted on a graph. The solution to the system of equations is the points where these lines intersect. If they intersect, the solution consists of the coordinates of the intersection point. If they do not intersect, like parallel lines, then there is no solution. In our example, converting both equations to slope-intercept form helps us determine whether the lines intersect or not.

Parallel Lines

Parallel lines are lines in a plane that never meet; they have the same slope but different y-intercepts. In the slope-intercept form, an equation looks like \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
Converting our given equations into this form:

• Equation 1: \( y = -\frac{7}{6}x - \frac{2}{3} \)
• Equation 2: \( y = -\frac{7}{6}x - \frac{2}{3} \)

Both equations share the same slope \( (-\frac{7}{6}) \) but have different y-intercepts \( (-\frac{2}{3}) \). This confirms that the lines are parallel. Parallel lines never touch each other, so they will not have any points in common on a graph.

No Solutions

In mathematics, a system of equations is said to have no solutions when there are no points that satisfy all equations involved. When dealing with linear equations, this occurs when the lines are parallel, as there are no intersection points.
In the context of our solved system, both lines were found to be parallel, indicated by their identical slopes and differing y-intercepts. Therefore, there are no x, y pairs that can satisfy both equations at once. This outcome is crucial because it tells us that trying to solve this system will not produce any viable solutions. In other words, there are no values of \( x \) and \( y \) that can simultaneously satisfy both equations. Recognizing this saves time and effort, allowing us to conclude the problem efficiently.