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Chapter 2: Problem 27

Each table of values gives several points that lie on a line. Write an equation in slope-intercept form of the line. $$ \begin{array}{|r|r|} \hline x & y \\ \hline-2 & -8 \\ \hline 0 & -4 \\ \hline 1 & -2 \\ \hline 3 & 2 \end{array} $$

Short Answer

Expert verified
y = 2x - 4

Step by step solution

01

Identify Points

From the table, identify the points given: Point 1: \(-2, -8\) Point 2: \(0, -4\) Point 3: \(1, -2\) Point 4: \(3, 2\)
02

Find the Slope

Use the slope formula, \m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\, to find the slope between two points (0, -4) and (1, -2): \[ m = \frac{{-2 - (-4)}}{{1 - 0}} = \frac{{-2 + 4}}{{1}} = 2 \]
03

Use Slope-Intercept Form

Use the slope-intercept form, \y = mx + b\. Substituting \m = 2\ and using point (0, -4) to find \b\: \[ -4 = 2(0) + b \] \[ -4 = b \]
04

Write the Equation

Substitute \m\ and \b\ into the slope-intercept form: \[ y = 2x - 4 \]

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Key Concepts

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

linear equations
A linear equation is a type of equation that represents a straight line on a graph. This kind of equation is pivotal in algebra and is typically written in the form \(y = mx + b\), known as the slope-intercept form. Each term in the equation has a specific part to play.
  • \(y\): the value on the y-axis (dependent variable).
  • \(x\): the value on the x-axis (independent variable).
  • \(m\): the slope of the line, showing how steep the line is.
  • \(b\): the y-intercept, the point where the line crosses the y-axis.
To fully understand how to write and manipulate these equations, knowing how to calculate the slope and intercept is essential. These concepts intertwine to help us graph linear equations and understand their behavior.
slope calculation
The slope \(m\) of a line indicates its steepness and is computed using two points on the line. The formula is
\[ m = \frac{(y2 - y1)}{(x2 - x1)} \]
where \((x1, y1)\) and \((x2, y2)\) are coordinates of two distinct points on the line. Slope determines how much the y-value changes for a given change in the x-value. Positive slope means the line rises, while negative slope means it falls.
  • Example Calculation: For points (0, -4) and (1, -2):
  • Change in \(y\): \(y2 - y1 = -2 - (-4) = -2 + 4 = 2\)
  • Change in \(x\): \(x2 - x1 = 1 - 0 = 1\)
  • Slope: \(m = \frac{2}{1} = 2\)
This means for each unit increase in \(x\), \(y\) increases by 2.
algebra
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. In the context of this problem, we use algebraic methods to solve for unknowns.

Slope-Intercept Form: Given the slope \(m\) and a point \((x, y)\), we can always find the y-intercept \(b\) by rearranging the equation to solve for \(b\). Using \(y = mx + b\):
  • Substitute known slope and point values.
  • Rearrange to solve for \(b\).
  • Example: With \(m = 2\) and point (0, -4):

    • \[ -4 = 2(0) + b \] \[ -4 = b \]

    This gives \(b = -4\). So, the equation becomes \(y = 2x - 4\). Once \(m\) and \(b\) are known, the linear equation can be written, providing a full description of its line on the graph.

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Most popular questions from this chapter

Let \(f(x)=-3 x+4\) and \(g(x)=-x^{2}+4 x+1 .\) Find the following $$ f(x+h) $$

Graph each line passing through the given point and having the given slope. (5,3)\(; m=0\)

A taxicab driver charges \(\$ 2.50\) per mile. (a) Fill in the table with the correct response for the price \(f(x)\) the driver charges for a trip of \(x\) miles. (b) The linear function that gives a rule for the amount charged is \(f(x)=\) (c) Graph this function for the domain \\{0,1,2,3\\} using the set of axes at the right. $$ \begin{array}{c|c} x & f(x) \\ \hline 0 & \\ \hline 1 & \\ \hline 2 & \\ \hline 3 & \\ \hline \end{array} $$

Find the slope of each line in three ways by doing the following. (a) Give any two points that lie on the line, and use them to determine the slope. See Example \(2 .\) (b) Solve the equation for \(y\), and identify the slope from the equation. See Example 4 . (c) For the form \(A x+B y=C,\) calculate \(-\frac{A}{B} .\) See the Note following Example 4 . \(2 x-y=8\)

Forensic scientists use the lengths of certain bones to calculate the height of a person. Two such bones are the tibia \((t),\) the bone from the ankle to the knee, and the femur \((r),\) the bone from the knee to the hip socket. A person's height \((h)\) in centimeters is determined from the lengths of these bones using the following functions. For men: \(\quad h(r)=69.09+2.24 r\) or \(\quad h(t)=81.69+2.39 t\) For women: \(\quad h(r)=61.41+2.32 r\) or \(h(t)=72.57+2.53 t\) (a) Find the height of a man with a femur measuring \(56 \mathrm{~cm}\). (b) Find the height of a man with a tibia measuring \(40 \mathrm{~cm} .\) (c) Find the height of a woman with a femur measuring \(50 \mathrm{~cm}\). (d) Find the height of a woman with a tibia measuring \(36 \mathrm{~cm}\).
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