Problem 44 - 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? $$ \left\\{\begin{aligned} -5 x+8 y &=8 \\ -5 x+8 y &=-28 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The system of equations has no solution.

Step 1: Convert equations to slope-intercept form

The standard form of a linear equation is \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. Given the two equations \(-5x + 8y = 8\) and \(-5x + 8y = -28\), we isolate \(y\) to find its slope-intercept form. For the first equation, add \(5x\) on each side and then divide by \(8\) to obtain \(y = 5/8x + 1\). For the second equation, add \(5x\) on each side and then divide by \(8\) to get \(y = 5/8x - 3.5\)

Step 2: Analyze the number of solutions based on characteristics of these equations

We can see from the form of these two lines that they have the same slope (\(m = 5/8\)) but different y-intercepts (\(1\) and \(-3.5\)). This means they are parallel lines. In a system of linear equations, parallel lines do not intersect each other, hence, the system has no solution.

Key Concepts

Linear Equations

A linear equation is any equation that can be written in the form \(y = mx + c\), where \(m\) and \(c\) are constants. This is known as the slope-intercept form. The equation represents a straight line when graphed. The slope \(m\) indicates the steepness of the line, and the y-intercept \(c\) shows where the line crosses the y-axis.
Linear equations are foundational in algebra and a powerful tool in many mathematical applications.Understanding the slope-intercept form is crucial because it allows us to easily identify and graph linear relationships.
For example, given the equation \(-5x + 8y = 8\), by rearranging to solve for \(y\), we find the equation \(y = \frac{5}{8}x + 1\). This represents a linear equation with a slope of \(\frac{5}{8}\) and a y-intercept of \(1\). This entire process helps in determining features of the line using just these two numbers.

Parallel Lines

Parallel lines are lines in the same plane that never intersect, regardless of how far they are extended. This is because parallel lines have the same slope.
In the context of linear equations, if two lines have identical slopes but different y-intercepts, they are parallel.Consider the equations from the problem statement: \(-5x + 8y = 8\) and \(-5x + 8y = -28\).
Converting both into slope-intercept form results in \(y = \frac{5}{8}x + 1\) and \(y = \frac{5}{8}x - 3.5\). Both lines have a slope of \(\frac{5}{8}\), confirming they are parallel.Since parallel lines do not meet, they do not have any points in common.
Therefore, if you're dealing with parallel lines in a system of equations, it directly influences the number of solutions possible.

System of Equations

A system of equations consists of two or more equations with the same set of variables. The goal is often to find values for the variables that satisfy all equations simultaneously.
Solutions to a system of equations can be visualized as the intersection points of the lines represented by the equations. There are three types of solutions for a system of linear equations:

• One solution: The lines intersect at one point, indicating one set of values work for both equations.
• No solution: The lines are parallel and never intersect.
• Infinitely many solutions: The lines overlap completely, meaning they are the same line repeated.

In our exercise, the system consists of two parallel lines: \(y = \frac{5}{8}x + 1\) and \(y = \frac{5}{8}x - 3.5\).
Since they never meet, there is no point that satisfies both equations simultaneously.

Number of Solutions

The number of solutions in a system of equations is determined by how the lines interact on a graph. This is crucial for understanding the outcomes you can expect from solving a set of equations.

• For one solution, the lines intersect at exactly one point. This gives a specific value for each variable.
• If there are no solutions, it means the lines are parallel and don’t intersect at all.
• An infinite number of solutions occurs when the equations represent the same line, overlapping completely.

In the given system: \(-5x + 8y = 8\) and \(-5x + 8y = -28\), the equations were converted to slope-intercept form.
The same slope but different y-intercepts confirm they are parallel lines and thus, result in no solutions. Understanding how to determine the number of solutions helps you predict the behavior of a system of equations.