Problem 49 - 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 \(f(x)=m x+b, f(-2)=11\) and \(f(3)=-9\). Find \(m\) and \(b\).

Step-by-Step Solution

Answer

The slope of the linear function is \(m = -4\) and the y-intercept is \(b = 3\)

Step 1: Ancillary Information

The linear function \(f(x) = mx + b\) is in slope-intercept form where \(m\) is the slope (rate of change) and \(b\) is the y-intercept (the value of \(y\) when \(x = 0\)). The given points \((-2, 11)\) and \((3, -9)\) can be substituted into the linear function to produce two linear equations. After substituting the points, we get two equations in the variables \(m\) and \(b\).

Step 2: Substitute the Points

If we substitute \((-2, 11)\) into \(f(x) = mx + b\), we get \(11 = -2m + b\). If we substitute \((3, -9)\) into \(f(x) = mx + b\), we get \(-9 = 3m + b\). Now we have a system of two linear equations in the variables \(m\) and \(b\).

Step 3: Solve the System

You can solve this system by the method of substitution or elimination, it depends on the comfort level of the student. If we subtract the second equation from the first we get: \(20 = -5m\)

Step 4: Find the Slope

Solving the above equation for \(m\) by dividing both sides by -5 gives \(m = -4\)

Step 5: Find the Intercept

Substituting \(m = -4\) into the equation \(11 = -2m + b\) gives \(11 = 8 + b\). Solving for \(b\) gives \(b = 3\)

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial when dealing with linear equations. The standard format is expressed as \( y = mx + b \), where \( m \) represents the slope, and \( b \) indicates the y-intercept, the point where the line crosses the y-axis.

In practical terms, the slope gives you the rate at which the y-coordinate changes for every one-unit increase in the x-coordinate. On the other hand, the y-intercept provides a starting point from which one can plot the line on a graph. For instance, in the problem given, the task is to identify the slope \( m \) and y-intercept \( b \) of the linear function \( f(x) = mx + b \) using the provided points.

Rate of Change

The rate of change is a fundamental concept in algebra that describes how one quantity changes in relation to another. In linear functions, the rate of change is consistent and is synonymous with the slope of the line, given by \( m \) in the slope-intercept equation.

To calculate the slope using two points, use the formula \( m = (y_2 - y_1)/(x_2 - x_1) \). This formula tells you how much the y-value (dependent variable) changes for a single change in the x-value (independent variable). The problem provides two points that allow you to find this constant rate of change for the function.

Substitution Method

The substitution method is one of the algebraic techniques used to solve systems of linear equations. It involves isolating one variable in one equation and then substituting that expression into the other equation. This method can make it easier to find a solution, particularly when one of the equations is already solved for a specific variable.

In the given example, the substitution method could be used after finding the value of the variable \( m \) by isolating it in one of the equations. Once \( m \) is known, it can be substituted back into either equation to find the value of \( b \).

Elimination Method

Another technique for solving systems of linear equations is the elimination method. This process involves adding or subtracting equations to eliminate one of the variables, making it possible to solve for the other variable.

For the exercise provided, the elimination method is applied by subtracting one of the equations from the other, ultimately removing the variable \( b \) and leaving a simple equation that can be solved for \( m \). After determining the slope, you can then proceed to find the y-intercept \( b \) by substituting the value of \( m \) back into one of the original equations.

Linear Equations

Linear equations form the foundation for understanding relationships between two variables. They graph as straight lines and have constant rates of change. An equation like \( y = mx + b \) is a classic example that describes a linear relationship. Each system of linear equations can have one solution, no solution, or infinitely many solutions.

In our specific case, we are given a system of two linear equations. Using algebraic techniques, we found one unique solution for the values of \( m \) and \( b \), illustrating that given two different points on a line, one can uniquely determine the linear equation that models that line.