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Chapter 1: Problem 14

Find the constants \(m\) and \(b\) in the linear function \(f(x)=\) \(m x+b\) such that \(f(2)=4\) and the straight line represented by \(f\) has slope \(-1\).

Short Answer

Expert verified
The constants in the linear function \(f(x) = mx + b\) are \(m = -1\) and \(b = 6\), making the function \(f(x) = -1x + 6\).

Step by step solution

01

Use given point in the function

We'll first use the point (2,4) in the function f(x) = mx + b. This will give us the equation: \(4 = m(2) + b\)
02

Solve equation for b

We can rewrite the equation from step 1 to solve for b: \(b = 4 - 2m\)
03

Use the given slope in the function

We're given that the slope of the line, which is equal to the value of m, is -1. So, we have: \(m = -1\)
04

Substitute m in the equation for b

Now, we can substitute the value of m from step 3 into the equation for b from step 2: \(b = 4 - 2(-1)\)
05

Solve for b

We can now solve for b: \(b = 4 + 2\) \(b = 6\)
06

Write the linear function

We now have the values for m and b, so we can write out the linear function: \(f(x) = -1x + 6 \)

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