Problem 50 - Thinking Mathematically

Question

Solve each system for \(x\) and \(y\), expressing either value in terms of a or b, if necessary. Assume that \(a \neq 0\) and \(b \neq 0\). For the linear function \(f(x)=m x+b, f(-3)=23\) and \(f(2)=-7\). Find \(m\) and \(b\).

Step-by-Step Solution

Answer

The slope (m) of the linear function is -6, and the y-intercept (b) is 5.

Step 1: Form the equations

First we know that \(f(x)\) is a function defined by \(mx + b\). We substitute the given \(x\) values from the problem into the function to produce two equations. This gives us the system of equations: \[m(-3) + b = 23\] and \[m(2) + b = -7\] or in a more simple form: \[-3m + b = 23\] and \[2m + b = -7\]

Step 2: Solve the system

We can solve this system by substitution or elimination. In this example, we'll use the method of elimination. First, we will subtract the second equation from the first. By subtracting the second equation from the first, the b terms are eliminated, making it easier to solve for m: \[(-3m + b) - (2m + b) = 23 - (-7)\]. Simplifying this gives: \[(-3m - 2m) + (b - b) = 30\], which results in \[-5m = 30\]. Solving for m, we get \(m = -6\)

Step 3: Find the y-intercept

Next, substitute the value of m we just found into either of the original equations to solve for b. For instance, using the first equation: \[-3(-6) + b = 23\], simplifying this yields \(18 + b = 23\), and solving for b, we get \( b = 5\)

Key Concepts

Systems of Equations

A system of equations is a set of equations with multiple variables. In general, it provides several equations that must all be true simultaneously. Solving a system means finding values for the variables that make all the equations true at the same time. In our exercise, we are dealing with two equations derived from a linear function. The goal is to solve for both variables, typically denoted as \(x\) and \(y\), but in our case, they are \(m\) and \(b\).

Such systems appear often in real-life problems where multiple conditions have to be satisfied together. For instance, if you want to find a point where two lines on a graph intersect, you would solve a system of equations representing those lines.

Solving by Elimination

The elimination method is one of the primary techniques for solving systems of equations. The basic idea is to add or subtract the equations to eliminate one of the variables. This simplifies the system and allows for an easier path to finding the solution.

In our exercise, we used elimination to subtract one equation from the other, which helped us eliminate the term \(b\). This reduction simplified our system to a single equation with one variable. We then easily solved for \(m\) first. Using elimination is particularly effective when dealing with equations that are already aligned in a way where one term can be cancelled out, as was the case here, where the \(b\) terms in both equations had the same coefficient.

Substitution Method

The substitution method is another powerful tool for solving systems of equations. This method involves solving one equation for one variable, then substituting this expression into the other equation. This effectively reduces the system from two equations with two variables to one equation and one unknown.

Although we didn't use substitution in our solution, it's a highly useful method, especially when one of the equations is already easy to solve for one variable. Imagine if our given equations let us express \(b\) in terms of \(m\) quickly. If that were the case, we could substitute back into the second equation to solve for \(m\), providing an alternative route to a solution.

Slope and Intercept

Slope and intercept are fundamental concepts in understanding linear functions. The equation of a line is often written in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

- **Slope \(m\):** This measures the steepness of a line. A positive slope means the line rises as it moves right, and a negative slope means it falls. In our exercise, after calculating, \(m\) turned out to be \(-6\), indicating a downward slope.
- **Y-Intercept \(b\):** This is the value of \(y\) at the point where the line crosses the y-axis. For the given linear function, we found \(b = 5\), so the line crosses the y-axis at \(y=5\).
Understanding these concepts aids in sketching the graph of the linear function and interpreting its behavior visually. This representation is crucial because it adds a visual dimension to the analytical solutions we pursue.