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Chapter 3: Problem 108

Assume that \(r, p,\) and \(s\) are constants and that \(x\) and \(y\) are variables. Determine the slope and the y-intercept. $$r x+p y=s-r y$$

Short Answer

Expert verified
Slope: \(\frac{-r}{p + r}\), y-intercept: \(\frac{s}{p + r}\)

Step by step solution

01

Combine like terms

First, move all the terms involving the variables to one side of the equation. Starting with the given equation:\[ r x + p y = s - r y \]Add \(r y\) to both sides to combine like terms:\[ r x + p y + r y = s \]This simplifies to:\[ r x + (p + r) y = s \]
02

Solve for y in terms of x

Rewrite the equation in the slope-intercept form \(y = mx + b\):\[ r x + (p + r) y = s \]Subtract \(r x\) from both sides:\[ (p + r) y = -r x + s \]Divide both sides by \(p + r\):\[ y = \frac{-r}{p + r} x + \frac{s}{p + r} \]
03

Identify the slope and the y-intercept

In the slope-intercept form \(y = mx + b\), the coefficient of \(x\) is the slope \(m\), and the constant term is the y-intercept \(b\). Thus, comparing:\[ y = \frac{-r}{p + r} x + \frac{s}{p + r} \]The slope \(m\) is \(\frac{-r}{p + r}\) and the y-intercept \(b\) is \(\frac{s}{p + r}\).

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Key Concepts

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

Linear Equations
Linear equations describe relationships between two variables that create a straight line when graphed. These equations typically come in the general form:

\[ ax + by = c \]
where
  • \(a\) and \(b\) are coefficients
  • \(x\) and \(y\) are variables
  • \(c\) is a constant
Linear equations are fundamental in algebra and are used to model real-world scenarios. The equation in the exercise \(r x + p y = s - r y\) can be rewritten to fit this linear form, which sets the stage for finding the slope and y-intercept.
Slope-Intercept Form
Slope-intercept form is a special way of writing linear equations, expressed as:

\[ y = mx + b \]
where
  • \(m\) is the slope
  • \(b\) is the y-intercept
This form makes it easy to identify these components directly from the equation. For example, after rearranging our original equation to
\[ y = \frac{-r}{p+r}x + \frac{s}{p+r} \]
we easily see
  • The slope \(m = \frac{-r}{p+r}\)
  • The y-intercept \(b = \frac{s}{p+r}\)
Understanding the slope and y-intercept helps in graphing the linear equation and analyzing the relationship between variables.
Solving for y
To transform any linear equation into slope-intercept form, you need to solve for \(y\). Let's break down our specific example step by step:
1. Start with the original equation:
\[r x + p y = s - r y \]
  • First, move all the \(y\)-terms to one side by adding \(r y\) to both sides:
    \[r x + p y + r y = s \]
  • Combine like terms:
    \[r x + (p + r) y = s \]
  • Solve for \(y\): First, isolate the y-term by subtracting \(r x\) from both sides:
    \[(p + r) y = -r x + s \]
  • Divide both sides by \(p + r\) to isolate \(y\):
    \[ y = \frac{-r}{p + r} x + \frac{s}{p + r} \]
Solving for \(y\) in this way allows you to write the equation in slope-intercept form \( y = mx + b \), making it straightforward to identify the slope and y-intercept.

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Most popular questions from this chapter

Write the slope-intercept equation of the line that has the same \(y\) -intercept as the line \(x-3 y=6\) and contains the point \((5,-1)\)

Find the function values. $$g(x)=2 x+3$$ a) \(g(0)\) b) \(g(-4)\) c) \(g(-7)\) d) \(g(8)\) e) \(g(a+2)\) f) \(g(a)+2\)

Pressure at Sea Depth. The pressure \(P(d),\) in atmospheres, at a depth \(d\) feet beneath the surface of the ocean is given by \(P(d)=0.03 d+1\). a) What do the numbers 0.03 and 1 signify? b) Where is the pressure 4 atmospheres? c) What is the domain of \(P ?\)

Find the indicated function values for each function. $$f(x)=\left\\{\begin{array}{ll}{x,} & {\text { if } x<0} \\\\{2 x+1,} & {\text { if } x \geq 0}\end{array}\right.$$ a) \(f(-5)\) b) \(f(0)\) c) \(f(10)\)

Use a graphing calculator to find the function values. $$p(a)=\frac{2}{a}-a^{2}$$ a) \(p\left(\frac{1}{8}\right)\) b) \(p(-0.5)\)
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