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Chapter 1: Problem 13

Find the constants \(m\) and \(b\) in the linear function \(f(x)=\) \(m x+b\) such that \(f(0)=2\) and \(f(3)=-1\).

Short Answer

Expert verified
The constants for the linear function are \(m = -1\) and \(b = 2\), so the function is \(f(x) = -x + 2\).

Step by step solution

01

Plug in the given points into the function

For the point \((0, 2)\), we have \(f(0) = 2\). So, we plug in \(x = 0\) and get the equation: \[2 = m(0) + b\] For the point \((3, -1)\), we have \(f(3) = -1\). So, we plug in \(x = 3\) and get the equation: \[-1 = m(3) + b\] Now, we have a system of two linear equations with two unknowns, \(m\) and \(b\): \begin{align*} 2 &= m(0) + b \\ -1 &= m(3) + b \end{align*}
02

Solve the system of equations for m and b

Start by solving the first equation for b: \[b = 2\] Now, substitute this value into the second equation: \[-1 = m(3) + 2\] Then, solve the equation for m: \begin{align*} -1 - 2 &= m(3) \\ -3 &= m(3) \\ m &= -1 \end{align*}
03

Write the linear function with the found values

Now that we have the values for \(m\) and \(b\), we can write the linear function \(f(x)\): \[f(x) = -1x + 2\] So, the linear function with the given points is \(f(x) = -x + 2\).

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