Problem 58 - Algebra Form and Function

Question

If \(b_{1}+m_{1} x=b_{2}+m_{2} x,\) what can be said about the constants \(b_{1}, m_{1}, b_{2},\) and \(m_{2}\) if the equation has (a) One solution? (b) No solutions? (c) An infinite number of solutions?

Step-by-Step Solution

Answer

Answer: For the given equation \(b_1 + m_1x = b_2 + m_2x\): - If the slopes \(m_1 \ne m_2\), the equation has one solution. - If the slopes \(m_1 = m_2\) and y-intercepts \(b_1 \ne b_2\), the equation has no solutions. - If the slopes \(m_1 = m_2\) and y-intercepts \(b_1 = b_2\), the equation has an infinite number of solutions.

Step 1: Case (a) One solution#

For the given equation to have one solution, the lines represented by the equation must intersect at exactly one point. This means the lines must not be parallel or coincident (coincident lines have an infinite number of intersection points). In order for two lines not to be parallel, their slopes should be different. So, for the given equation \(b_1 + m_1x = b_2 + m_2x\) to have one solution, their slopes \(m_1\) and \(m_2\) must be different: \(m_1 \ne m_2\) In this case, there are no constraints on the values of \(b_1\) and \(b_2\).

Step 2: Case (b) No solutions#

For the given equation to have no solutions, the lines represented by the equation must be parallel and not coincident. In order for two lines to be parallel, their slopes should be equal. So, for the given equation \(b_1 + m_1x = b_2 + m_2x\) to have no solutions, their slopes \(m_1\) and \(m_2\) must be equal: \(m_1 = m_2\) Also, we need these parallel lines to never intersect. In this case, their y-intercepts should be different: \(b_1 \ne b_2\)

Step 3: Case (c) An infinite number of solutions#

For the given equation to have an infinite number of solutions, the lines represented by the equation must be coincident, meaning they are the same line. In order for two lines to be coincident, their slopes and y-intercepts should be equal. So, for the given equation \(b_1 + m_1x = b_2 + m_2x\) to have an infinite number of solutions, the following conditions must hold: \(m_1 = m_2\) and \(b_1 = b_2\) In summary, for the given equation \(b_1 + m_1x = b_2 + m_2x\): - If the slopes \(m_1 \ne m_2\), the equation has one solution. - If the slopes \(m_1 = m_2\) and y-intercepts \(b_1 \ne b_2\), the equation has no solutions. - If the slopes \(m_1 = m_2\) and y-intercepts \(b_1 = b_2\), the equation has an infinite number of solutions.

Key Concepts

Slope in Linear Equations

The **slope** of a line is a measure of how steep the line is. It is represented by the letter **m** in linear equations of the form \( y = mx + b \). The slope can tell us a lot about the behavior of lines on a graph:

• **Positive slope**: The line rises as it moves from left to right.
• **Negative slope**: The line falls as it moves from left to right.
• **Zero slope**: The line is horizontal, indicating no rise or fall.
• **Undefined slope**: The line is vertical.

In the context of equations like \( b_1 + m_1x = b_2 + m_2x \), knowing the values of \( m_1 \) and **m_2** helps determine if the lines are parallel, coincident, or if they intersect at some point. When exploring different solutions for the equations, differences or similarities in slopes determine the nature of their intersections.

Parallel Lines in Mathematics

When we say two lines are **parallel**, this means that they have the same slope and will never meet, no matter how far they are extended. In terms of linear equations, if you have two lines, and they are represented by equations like \( y = m_1x + b_1 \) and \( y = m_2x + b_2 \), then for the lines to be parallel, we need:

• **Equal slopes**: \( m_1 = m_2 \)
• **Different y-intercepts**: \( b_1 eq b_2 \)

This means that despite having the same direction, parallel lines never touch or cross each other. This uniqueness of parallel lines is important when solving systems of equations, as it is essential to recognize when a system has no solutions just because the lines never intersect.

Coincident Lines and Infinite Solutions

**Coincident lines** occur when two lines lie exactly on top of each other on a graph. This situation is special because it means that the lines are not just similar but identical in position. For lines to be coincident, we require:

• **Identical slopes**: \( m_1 = m_2 \)
• **Same y-intercepts**: \( b_1 = b_2 \)

Since they are the same line, any point on one line is also a point on the other line, leading to an **infinite number of solutions**. This is because any value of \( x \) you choose will satisfy both equations, indicating the lines overlap perfectly. Understanding coincident lines is crucial when discussing systems of linear equations, as it speaks to the concept of infinite solutions, highlighting complete agreement between the equations.