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Chapter 2: Problem 82

Use the two given functions to write \(y\) as a function of \(x\). $$y=-4 d-1, d=-3 x-2$$

Short Answer

Expert verified
y = 12x + 7

Step by step solution

01

Identify the given functions

The problem provides two functions: 1. Function for y: y = -4d - 1y = f(d)2. Function for d: d = -3x - 2d = g(x)
02

Substitute the value of d in the function for y

To find y as a function of x, substitute the function for d into the function for y: y = -4(-3x - 2) - 1
03

Simplify the equation

Distribute -4 to both terms inside the parentheses: y = -4(-3x) - 4(-2) - 1 y = 12x + 8 - 1
04

Combine like terms

Combine the constant terms to simplify: y = 12x + 7

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution Method
The substitution method involves replacing a variable in one equation with an expression from another equation.
In our given problem, we have two functions:
\( y = -4d - 1 \) and \( d = -3x - 2 \)
We want to express \( y \) as a function of \( x \).
To do this, we substitute the expression for \( d \) from the second function into the first function.
So we replace \( d \) in \( y = -4d - 1 \) with \( -3x - 2 \). This gives us: \[ y = -4(-3x - 2) - 1 \]
This substitution helps us eliminate \( d \), allowing us to write \( y \) directly in terms of \( x \).
Algebraic Manipulation
In the process of solving equations, algebraic manipulation is essential.
It involves simplifying expressions, expanding products, and combining like terms.
After substituting \( d = -3x - 2 \) into \( y = -4d - 1 \), we get:
\( y = -4(-3x - 2) - 1 \)
Next, we distribute the \( -4 \) to both \( -3x \) and \( -2 \):
\[ y = -4(-3x) - 4(-2) - 1 \]
Multiplying these out, we get:
\[ y = 12x + 8 - 1 \]
Properly applying these multiplication and distribution rules ensure we get the right terms to further simplify.
Simplifying Expressions
The final crucial step in solving our function composition problem is simplifying the expression.
Combining like terms helps reduce the equation to its simplest form.
After distributing \( -4 \), we have:
\[ y = 12x + 8 - 1 \]
Here, \( 8 \) and \( -1 \) are constant terms.
We can simplify further by combining these constants:
\[ y = 12x + 7 \]
This is the final, simplified expression of \( y \) in terms of \( x \).
By performing these simplifications, we make the function easier to understand and use.

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