Problem 51 - College Algebra Essentials

Question

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 \(m\) is -4 and the y-intercept \(b\) is 3.

Step 1: Substitute the given values into the function

The given function is \(f(x) = mx + b\), so substitute \(x = -2\) into the function to get \(11 = m(-2) + b\), which simplifies to \(11 = -2m + b\). Similarly, substituting \(x = 3\) into the function, yields \(-9 = 3m + b\).

Step 2: Set up a system of equations

From the previous step, two equations are obtained: \(11 = -2m + b\) and \(-9 = 3m + b\). These form a system of equations that can be used to solve for \(m\) and \(b\).

Step 3: Solve the system of equations

Subtract the second equation from the first: \(11 - (-9) = -2m - 3m + b - b\). This simplifies to \(20 = -5m\). Dividing by -5 provides \(m = -4\). Substitute \(m = -4\) into the first equation to solve for \(b\). This gives \(11 = 8 + b\). Subtracting 8 from both sides provides \(b = 3\).

Key Concepts

Understanding Systems of Equations

When dealing with a system of equations, you are essentially working with two or more equations that share common variables. In the example above, we encountered two equations:

• \(11 = -2m + b\)
• \(-9 = 3m + b\)

These equations form a system because they both involve the same variables, \(m\) and \(b\). The goal is to find values for these variables that satisfy both equations simultaneously.

Solving systems of equations is crucial because many real-world problems require finding multiple unknowns. It helps to visualize this by thinking of each equation as a line on a graph. The solution to the system is where these lines intersect. In our example, we found the point where both equations met by solving for \(m\) and \(b\).

Introducing Slope-Intercept Form

The slope-intercept form of a linear equation is particularly user-friendly because it provides a straightforward way to identify the slope and y-intercept of a line. In its standard form, it is written as:

• \(f(x) = mx + b\)

where \(m\) represents the slope, and \(b\) represents the y-intercept of the line.

Understanding this form is powerful because it immediately tells you how the line behaves. The slope \(m\) indicates how steep the line is and in what direction it goes. A positive slope means the line rises, while a negative slope means it falls as you move along the x-axis. The y-intercept \(b\) tells you the point where the line crosses the y-axis.

In our example, transforming given conditions into this form allowed us to extract critical pieces of information to solve for unknowns \(m\) and \(b\).

Solving Linear Equations Efficiently

To solve linear equations, especially in a system, there are several methods that you can apply. The substitution method, shown in the example, is one of the most intuitive. It involves substituting one equation into another to eliminate a variable and solve for the remaining one.

Here’s a simplified strategy for solving such equations:

• Solve one of the equations for one of the variables if not already done.
• Substitute this expression into the other equation.
• Solve the resulting single-variable equation.
• Use the found value to solve for the other variable.

In our step-by-step solution, after setting our equations, we subtracted one from the other to eliminate \(b\) and find \(m = -4\). With \(m\) known, substituting back into any of the original equations yields \(b = 3\). This method ensures clarity and reduces the high chances of error in complex calculations.