Problem 51 - Algebra and Trigonometry

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 values of \(m\) and \(b\) are -4 and 19 respectively.

Step 1: Substitute \(x = -2\) in the equation of the line

Substitute \(x = -2\) in the equation \(f(x) = mx + b\). This gives us the equation \(11 = -2m + b\).

Step 2: Substitute \(x = 3\) in the equation of the line

Substitute \(x = 3\) in the equation \(f(x) = mx + b\). This leads to the equation \(-9 = 3m + b\).

Step 3: Solve the system of linear equations

We now have a system of two linear equations in two variables, \(-2m + b = 11\) and \(3m + b = -9\). Subtract the second equation from the first to eliminate \(b\) and solve for \(m\). This yields \(m= -4\). Substitute \(m = -4\) in one of the two equations to find \(b\). This leads to \(b = 19\).

Key Concepts

System of Equations

In mathematics, a system of equations is a collection of two or more equations with a common set of variables. When dealing with linear functions, such a system often includes equations that describe straight lines. A solution to a system of equations is where the graphs of the equations intersect, meaning it is a point that satisfies all the equations simultaneously.
For example, in the exercise provided, we have two equations derived from the linear function:

• \(11 = -2m + b\)
• \(-9 = 3m + b\)

These two equations form a system of linear equations because they both involve the same two variables, \(m\) and \(b\). The task is to find values for these variables that satisfy both equations at the same time. This is essential in finding the correct slope \(m\) and intercept \(b\) of the line described by the initially given function.

Solving for Variables

Solving for variables in a system of equations involves determining the specific values for these variables that satisfy all equations in the system. This process essentially means pinpointing where the graphs intersect.
Given the exercise details, the equations \(11 = -2m + b\) and \(-9 = 3m + b\) need to be solved for \(m\) and \(b\). Usually, we can tackle such equations using methods like substitution or elimination to simplify and find the required values:

• By elimination, as shown in the solution, one of the variables can be eliminated by subtracting one equation from the other.

The elimination helps isolate one variable, allowing us to substitute back into one of the original equations to find the second variable.

Substitution Method

The substitution method is a technique for solving a system of equations where one equation is solved for one variable in terms of the others, and this expression is then substituted back into the other equation(s). This helps in finding the solution by focusing on a single equation after substitution.
Let's explore how substitution could work with our example equations:

• Solve \(11 = -2m + b\) for \(b\), getting \(b = 11 + 2m\).
• Substitute this expression for \(b\) into the other equation \(-9 = 3m + b\), resulting in the equation \(-9 = 3m + (11 + 2m)\).

By solving this modified equation, you can determine \(m\), which can be substituted back into \(b = 11 + 2m\) to find the corresponding \(b\). The substitution method is straightforward and illustrative for systems with fewer variables.

Equation of a Line

The equation of a line in two-dimensional space can be written in the form \(f(x) = mx + b\), where \(m\) represents the slope of the line and \(b\) is the y-intercept. This equation shows the relationship between \(x\) and \(f(x)\), providing a graphical representation as a straight line.
The slope \(m\) indicates how steep the line is. A positive slope means the line rises as \(x\) increases, while a negative slope means it falls. In the exercise, the calculated slope \(m = -4\) indicates a downward slope. The y-intercept \(b\) is the point where the line crosses the y-axis. A value of \(b = 19\) suggests the line intersects the y-axis at \(19\).
Understanding these components helps in graphing the line and analyzing its properties, such as intercepts and steepness, through the function's formula.