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Chapter 1: Problem 52

Finding Equations of Lines and Graphing (a) Sketch the line with slope \(-2\) that passes through the point \((4,-1)\) (b) Find an equation for this line.

Short Answer

Expert verified
The equation of the line is \( y = -2x + 7 \).

Step by step solution

01

Understanding the Slope-Intercept Form

To find the equation of a line, we start by using the slope-intercept form of a line, which is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
02

Using Given Slope and Point

We are given a slope \( m = -2 \) and a point \((x_1, y_1) = (4, -1)\). We can use the point to solve for \( b \) in the equation \( y = mx + b \).
03

Substitute to Find the Y-Intercept

Plug the point \((4, -1)\) and slope \(-2\) into the slope-intercept form: \( -1 = -2(4) + b \). Simplifying, we get \( -1 = -8 + b \).
04

Solve for the Y-Intercept

To solve for \( b \), add 8 to both sides: \( -1 + 8 = b \) gives us \( b = 7 \).
05

Write the Equation of the Line

Now that we have both the slope and the y-intercept, the equation of the line is \( y = -2x + 7 \).
06

Sketch the Line

To sketch the line, plot the y-intercept \( (0, 7) \) on the graph. From this point, use the slope \(-2\) (which corresponds to a rise of -2 and run of 1) to find another point on the line. This can be repeated to confirm the line's direction. Draw the line through these points.

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form is a fundamental concept in linear equations. It expresses a line's equation as \( y = mx + b \), where:
  • \( m \) represents the slope of the line, indicating the steepness and direction.
  • \( b \) stands for the y-intercept, the point where the line crosses the y-axis.
This form is particularly useful because it allows us easily to write an equation once we know a line's slope and y-intercept. The formula helps to quickly graph a line and understand its behavior. The slope \( m \) tells us how much the y-value changes for a one-unit increase in the x-value. Meanwhile, the y-intercept \( b \) tells us where the line will intersect with the y-axis. Utilizing the slope-intercept form saves time when analyzing or graphing linear equations.
Graphing Linear Equations
Graphing linear equations can be enjoyable and straightforward when using the slope-intercept form. Begin by identifying the y-intercept, which is given by \( b \) in the formula \( y = mx + b \). This point tells us where to start plotting on the y-axis.
With the y-intercept marked, use the slope \( m \) to determine the line's direction. For example, if \( m = -2 \), this means for every step right (positive x-direction), you move two steps down (negative y-direction).
  • If \( m \) is positive, the line ascends; a negative \( m \) means it descends.
  • A zero slope indicates a horizontal line, while an undefined slope reflects a vertical line.
Combine several points by continuously applying the slope from the y-intercept for a precise and accurate line graph.
Finding the Y-Intercept
Determining the y-intercept is a pivotal step in finding a line's equation. With a known point and slope, the formula \( y = mx + b \) makes this task easy. Given a point, such as \((4, -1)\), and a slope \( m = -2 \), insert these into the slope-intercept equation. Start by addressing the equation's left side with the y-coordinate from the point: \(-1 = -2(4) + b\).
By solving this equation step-by-step, first compute \(-2 \times 4 = -8\). Then, solve for \( b \) by adding 8 to both sides, giving you \( b = 7 \).
This calculation indicates that the line's y-intercept is at \( (0, 7) \), essential for both graphing and writing the complete equation of the line.
Point-Slope Form
The point-slope form is another valuable equation form, particularly when you have a point on the line and its slope. This form is expressed as \( y - y_1 = m(x - x_1) \), where:
  • \((x_1, y_1)\) represents the coordinates of a known point on the line.
  • \( m \) denotes the slope of the line.
This form is particularly handy in scenarios where the y-intercept isn't readily apparent. Using our example of slope \(-2\) and point \((4, -1)\), substitute into the point-slope form: \( y + 1 = -2(x - 4) \).
Expand and simplify to find a different representation of the line's equation. Transitioning to slope-intercept or standard form is a viable next step for further calculations or graphing tasks. This flexibility is part of what makes point-slope so useful in understanding linear relationships.

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Most popular questions from this chapter

DISCUSS: Proof That \(0=1 ?\) The following steps appear to give equivalent equations, which seem to prove that \(1=0 .\) Find the error. \(\begin{aligned} x &=1 & & \text { Given } \\ x^{2} &=x & & \text { Multiply by } x \\ x^{2}-x &=0 & & \text { Subtract } x \end{aligned}$$(x-1)=0\)Factor \(\frac{(x-1)}{x-1}=\frac{0}{x-1} \quad\) Divide by \(x-1$$x=0 \quad\) Simplify\(1=0 \quad\) Given \(x=1\)

Use scientific notation, the Laws of Exponents, and a calculator to perform the indicated operations. State your answer rounded to the number of significant digits indicated by the given data. $$\frac{(73.1)\left(1.6341 \times 10^{28}\right)}{0.0000000019}$$

Manufacturer's Profit If a manufacturer sells \(x\) units of a certain product, revenue \(R\) and cost \(C\) (in dollars) are given by $$ \begin{array}{l} R=20 x \\ C=2000+8 x+0.0025 x^{2} \end{array} $$ Use the fact that profit \(=\) revenue \(-\) cost to determine how many units the manufacturer should sell to enjoy a profit of at least \(\$ 2400\).

Simplify the expression. (a) \(32^{2 / 5}\) (b) \(\left(\frac{4}{9}\right)^{-1 / 2}\) (c) \(\left(\frac{16}{81}\right)^{3 / 4}\)

DISCOVER - PROVE: Relationship Between Solutions and Coefficients The Quadratic Formula gives us the solutions of a quadratic equation from its coefficients. We can also obtain the coefficients from the solutions. (a) Find the solutions of the equation \(x^{2}-9 x+20=0\) and show that the product of the solutions is the constant term 20 and the sum of the solutions is \(9,\) the negative of the coefficient of \(x\) (b) Show that the same relationship between solutions and coefficients holds for the following equations:$$ \begin{array}{l}x^{2}-2 x-8=0 \\\x^{2}+4 x+2=0\end{array}$$ (c) Use the Quadratic Formula to prove that in general, if the equation \(x^{2}+b x+c=0\) has solutions \(r_{1}\) and \(r_{2}\) then \(c=r_{1} r_{2}\) and \(b=-\left(r_{1}+r_{2}\right)\)
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