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Chapter 7: Problem 30

Determine the slope and \(y\) -intercept of the lines. $$ y=3 x-11 $$

Short Answer

Expert verified
Answer: The slope of the line is \(3\) and the y-intercept is \(-11\).

Step by step solution

01

Identify the slope-intercept form in the given equation

The given equation is \(y = 3x - 11\). Comparing this equation with the slope-intercept form (\(y = mx + b\)), we can see that it is already in the required form.
02

Identify the slope (m) and the y-intercept (b)

Since our equation is \(y = 3x - 11\), looking at the coefficients, we can identify the slope (m) and the y-intercept (b): - The slope (m) is the coefficient of x which is \(3\). - The y-intercept (b) is the constant term which is \(-11\).
03

Write down the solution

The slope of the line is \(m = 3\), and the y-intercept is \(b = -11\).

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Key Concepts

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

Linear Equations
Linear equations are equations that form a straight line when graphed on a coordinate plane. They are typically written in the form of \(y = mx + b\), also known as the slope-intercept form.
  • Standard Form: Linear equations can sometimes appear in different forms, like \(Ax + By = C\), but transforming them into the slope-intercept form reveals the slope and y-intercept more clearly.
  • Graphing Made Easy: Once in slope-intercept form, plotting the equation becomes straightforward since you can quickly determine the slope and the y-intercept.

Linear equations are foundational in algebra as they help describe relationships between variables.You often encounter them in real-world scenarios, like predicting expenses or calculating speeds.Understanding how to manipulate and interpret them is crucial in solving many mathematical problems.
Slope
The slope of a line represents how steep the line is. It tells us how much \(y\) changes for a change in \(x\). In the equation \(y = mx + b\), the slope is represented by \(m\).
  • Positive Slope: If \(m > 0\), the line rises as you move from left to right, indicating a positive relationship between \(x\) and \(y\).
  • Negative Slope: If \(m < 0\), the line descends as you move from left to right, showing a negative relationship.
  • Zero Slope: A zero slope means the line is horizontal, showing no change in \(y\) as \(x\) changes.

The slope is a crucial concept for understanding how two quantities are related.In our exercise, the slope is 3, indicating a steep rise in the line, with \(y\) increasing by 3 units for every 1 unit \(x\) increases.
Y-Intercept
The y-intercept is the point where the line crosses the y-axis. It occurs when \(x = 0\) and is denoted as \(b\) in the slope-intercept form \(y = mx + b\).
  • Interpreting the Y-Intercept: The y-intercept shows the value of \(y\) when \(x\) is zero. It acts as the starting value for the line, before any changes in \(x\) are considered.
  • Graphical Clarity: On a graph, finding the y-intercept is as simple as locating where the line crosses the y-axis.
  • Real-Life Application: In practical situations, the y-intercept can represent initial costs, starting quantities, or other baseline measurements before any changes occur.

In the example provided, the y-intercept is -11.This means that when \(x = 0\), the value of \(y\) is -11.The y-intercept gives us important information about the line before its slope takes effect.

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Most popular questions from this chapter

Graph the inequalities. $$ y \geq-2 $$

Find the slope, if it exists, of the line through the given pairs of points. $$ (8,-2), \quad(10,-6) $$

Find the slope, if it exists, of the line through the given pairs of points. $$ (5,2), \quad(6,3) $$

For the following problems, write the equation of the line using the given information in slope-intercept form. $$ (1,6),(-1,-6) $$

For the following problems, find the equation of the line using the information provided. Write the equation in slope-intercept form. slope \(=\frac{2}{3}, \quad\) passes through (-1,2)
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