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Chapter 2: Problem 50

Write each equation in slope鈥搃ntercept form. Then find the slope and the y-intercept of the line determined by the equation. $$ 5(2 x-3 y)=4 $$

Short Answer

Expert verified
Slope: \(\frac{2}{3}\), Y-intercept: \(-\frac{4}{15}\).

Step by step solution

01

Distribute and Simplify

Start by expanding the equation using the distributive property: \[ 5(2x - 3y) = 4 \] This simplifies to: \[ 10x - 15y = 4 \]
02

Isolate y-Term

We need to isolate the term with \(y\) on one side. Subtract \(10x\) from both sides: \[ -15y = -10x + 4 \]
03

Solve for y

Divide every term by \(-15\) to solve for \(y\): \[ y = \frac{-10x}{-15} + \frac{4}{-15} \] Simplify the fractions: \[ y = \frac{2}{3}x - \frac{4}{15} \]
04

Identify Slope and Y-Intercept

In the equation \(y = \frac{2}{3}x - \frac{4}{15}\), the slope \(m\) is \(\frac{2}{3}\) and the y-intercept \(b\) is \(-\frac{4}{15}\).

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Key Concepts

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

Understanding Linear Equations
Linear equations are fundamental in mathematics. These equations showcase a relationship between two variables. Generally, they are expressed in the form \(ax + by = c\). By rearranging or manipulating these equations, we can analyze various characteristics such as slope and intercepts. Their straightforward nature makes them a common tool in algebra.Some key points about linear equations:
  • Linear equations will graph as straight lines. No curves, just one consistent slope.
  • They allow us to predict one variable based on the other.
  • They're essential in fields like physics, economics, and beyond for modeling relationships.
For our exercise, we transform the given linear equation into slope鈥搃ntercept form to easily identify its properties.
The Slope: What It Represents
The slope in a linear equation provides the measure of steepness or the rate at which one variable changes relative to the other. In the slope鈥搃ntercept form \(y = mx + b\), the slope is represented by \(m\).Here's more about slopes:
  • A positive slope means the line is rising as it moves from left to right.
  • A negative slope indicates the line is falling as it moves from left to right.
  • If the slope is zero, the line is horizontal, indicating no vertical change.
For the solved equation \(y = \frac{2}{3}x - \frac{4}{15}\), the slope \(m\) is \(\frac{2}{3}\), suggesting an upward climb as \(x\) increases.Understanding the slope is crucial in identifying how variables are interconnected over a graph.
Unveiling the Y-Intercept
The y-intercept is where the line crosses the y-axis, and it gives a starting point for the graph. In the slope鈥搃ntercept form, this is the \(b\) in \(y = mx + b\).Important insights about the y-intercept:
  • It shows the value of \(y\) when \(x\) is zero.
  • It provides an anchor point, helping to establish the graph's position.
  • Similarly to the slope, the y-intercept is crucial in sketching a line graph quickly.
In our equation \(y = \frac{2}{3}x - \frac{4}{15}\), the y-intercept is \(-\frac{4}{15}\), meaning the line crosses the y-axis at this point.Overall, the y-intercept is a simple yet powerful tool in graphing and understanding linear relationships.

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Most popular questions from this chapter

Let \(f(x)=\frac{3}{2} x-2\) For what value of \(x\) does function \(f\) have the given value? \(f(x)=\frac{2}{3}\)

Find the slope of the line that passes through \((a, b)\) and \((-b,-a) .\) Assume \(a \neq 0\) and \(b \neq 0\)

Let \(f(x)=-2 x+5 .\) For what value of \(x\) does function \(f\) have the given value? \(f(x)=-7\)

Complete table. \(g(a)=a^{3}\) \(\begin{array}{|c|c|}\hline \text { Input } & \text { Output } \\\\\hline-\frac{3}{4} & \\\\\frac{1}{6} & \\\\\frac{5}{2} & \\\\\hline\end{array}\)

Complete table. \(g(b)=2\left(-b-\frac{1}{4}\right)\) \(\begin{array}{|r|l|}\hline \boldsymbol{b} & \boldsymbol{g}(\boldsymbol{b}) \\\\\hline-\frac{3}{4} & \\\\\frac{1}{6} & \\\\\frac{5}{2} & \\\\\hline\end{array}\)
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