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Chapter 1: Problem 109

To what linear function of \(x\) does the linear equation \(a x+b y=c(b \neq 0)\) correspond? Why did we specify \(b \neq 0 ?\)

Short Answer

Expert verified
The linear function of \(x\) corresponding to the linear equation \(ax + by = c\) is \(y = -\frac{a}{b}x + \frac{c}{b}\), and the constraint \(b \neq 0\) is given to ensure that the equation remains linear and valid.

Step by step solution

01

1. Identify the linear equation

The given linear equation is \(a x+b y = c\), where \(b \neq 0\).
02

2. Solve for y in terms of x

To find the linear function of \(x\), we need to isolate \(y\) on one side of the equation. We can do this by subtracting \(ax\) from both sides and then dividing by \(b\): \[y = \frac{c - ax}{b}\]
03

3. Simplify the linear function

We can rewrite the linear function as: \[y = -\frac{a}{b}x + \frac{c}{b}\]
04

4. Identify the condition \(b \neq 0\)

The reason why it's specified that \(b \neq 0\) is because if \(b\) was equal to 0, the equation would become invalid when dividing by \(b\). Moreover, if \(b=0\), the equation would no longer be linear in terms of \(x\) and \(y\). In conclusion, the linear function of \(x\) corresponding to the given linear equation is \(y = -\frac{a}{b}x + \frac{c}{b}\), and the constraint \(b \neq 0\) is given to ensure that the equation remains linear and valid.

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