For the linear function f(x) = mx + b, f(−2) = 11 and ƒ(3) = -9. Find m and b.

Verified step by step guidance

1. Start with the general form of the linear function: $f\left(x\right)=mx+b$, where $m$ is the slope and $b$ is the y-intercept.
2. Use the given values to create two equations by substituting the points into the function. For $f(-2)=11$, substitute $x=\; -2$ and $f(-2)\; =\; 11$ to get: $m(-2)\; +\; b\; =\; 11$.
3. Similarly, for $f\left(3\right)=-9$, substitute $x=\; 3$ and $f(3)\; =\; -9$ to get: $m(3)\; +\; b\; =\; -9$.
4. Now you have a system of two linear equations: $-2m\; +\; b\; =\; 11$ and $3m\; +\; b\; =\; -9$. Use either the substitution or elimination method to solve for $m$ and $b$.
5. After solving the system, you will find the values of $m$ and $b$ that satisfy both equations, which define the linear function.

Key Concepts

Linear Functions

A linear function is an equation of the form f(x) = mx + b, where m represents the slope and b the y-intercept. It produces a straight line when graphed, and each input x corresponds to exactly one output f(x). Understanding this form is essential for identifying and working with linear relationships.

Slope of a Line

The slope (m) measures the rate of change of the function, indicating how much f(x) changes for a unit change in x. It can be found using two points (x₁, y₁) and (x₂, y₂) with the formula m = (y₂ - y₁) / (x₂ - x₁). Calculating the slope is crucial for determining the linear function's behavior.

Solving Systems of Equations

Given two points on a line, you can create two equations by substituting the points into f(x) = mx + b. Solving this system simultaneously allows you to find the values of m and b. Techniques include substitution or elimination, which are fundamental for solving multiple-variable problems.