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Chapter 1: Problem 31

In Problems \(29-34\), find an equation for each line. Then write your answer in the form \(A x+B y+C=0\). With \(y\)-intercept 3 and slope 2

Short Answer

Expert verified
The standard form is \(2x - y = -3\).

Step by step solution

01

Identify the Given Information

We are asked to find an equation of a line with a given slope and y-intercept. The slope (m) is 2 and the y-intercept (c) is 3.
02

Formulate the Slope-Intercept Equation

Recall the formula for the slope-intercept form of a line: \[ y = mx + c \]Substitute the given slope and y-intercept: \[ y = 2x + 3 \]
03

Convert to Standard Form

We need to convert the slope-intercept form to the standard form \(Ax + By + C = 0\). Start by rearranging:\[ y = 2x + 3 \]Subtract 2x from both sides:\[ -2x + y = 3 \]
04

Finalize the Standard Form

The equation \(-2x + y = 3\) isn't quite in standard form since A should be positive. Multiply the whole equation by -1:\[ 2x - y = -3 \]

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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 one of the most commonly used ways to write the equation of a line in algebra. The general format of this equation is \( y = mx + c \). Here, \( m \) represents the slope of the line, and \( c \) stands for the y-intercept. This form is particularly useful for quickly identifying how a line behaves on a graph. To better grasp this concept, let's consider:
  • **Slope (\( m \)):** It determines the steepness and the direction of the line. If \( m \) is positive, the line slopes upwards; if negative, it slopes downwards.
  • **Y-Intercept (\( c \)):** This is where the line crosses the y-axis, indicating the value of \( y \) when \( x \) is 0.
By using this form, one can quickly sketch the graph of a line just by knowing the slope and the y-intercept.
standard form
The standard form of a linear equation is another essential format, often expressed as \( Ax + By + C = 0 \). In this equation:
  • \( A \), \( B \), and \( C \) are integers, with \( A \) typically kept positive.
  • Both \( A \) and \( B \) cannot be zero simultaneously because this would not define a line.
The standard form is beneficial because it is versatile and can be used to find intercepts easily:
  • To find the x-intercept, set \( y = 0 \) and solve for \( x \).
  • Conversely, to find the y-intercept, set \( x = 0 \) and solve for \( y \).
Converting from the slope-intercept form to the standard form often involves rearranging terms and sometimes multiplying through by -1 to ensure \( A \) is positive, making the equation adhere to conventional standards.
y-intercept
The y-intercept is a fundamental part of linear equations, as it indicates the point at which a line crosses the y-axis of a graph. It is denoted by \( c \) in the slope-intercept form \( y = mx + c \). Understanding the y-intercept can simplify graphing and interpreting linear relationships:
  • **Visual Interpretation:** If you plot the line on a graph, you would place a point on the y-axis at \( y = c \). This is your starting point when drawing the line.
  • **Equation Insight:** The y-intercept helps you see how changes in \( x \) affect \( y \) when the line crosses the y-axis.
The simplicity of identifying a y-intercept makes it appealing for quickly sketching graphs, teaching slope dynamics, and understanding the linearity of two related variables.
slope
The slope of a line is a crucial concept in algebra, contributing significantly to understanding how a line behaves on a graph. Denoted as \( m \) in the equation \( y = mx + c \), the slope indicates the rate at which \( y \) changes as \( x \) changes. It is often referred to as the line's 'steepness'. Here are some key details about slope:
  • **Positive Slope:** If \( m > 0 \), the line inclines upward from left to right.
  • **Negative Slope:** If \( m < 0 \), the line declines downward from left to right.
  • **Zero Slope:** If \( m = 0 \), the line is horizontal, indicating no change in \( y \) regardless of changes in \( x \).
  • **Undefined Slope:** Vertical lines have an undefined slope, as the change in \( x \) is zero.Understanding slope helps in predicting how a line extends as \( x \) values increase or decrease, which is pivotal in many applications such as physics and economics.

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Most popular questions from this chapter

In Problems 35-38, find the slope and \(y\)-intercept of each line. \(3 y=-2 x+1\)

Which of the following represent the same graph? Check your result analytically using trigonometric identities. (a) \(y=\sin \left(x+\frac{\pi}{2}\right)\) (b) \(y=\cos \left(x+\frac{\pi}{2}\right)\) (c) \(y=-\sin (x+\pi)\) (d) \(y=\cos (x-\pi)\) (e) \(y=-\sin (\pi-x)\) (f) \(y=\cos \left(x-\frac{\pi}{2}\right)\) (g) \(y=-\cos (\pi-x)\) (h) \(y=\sin \left(x-\frac{\pi}{2}\right)\)

We now explore the relationship between \(A \sin (\omega t)+\) \(B \cos (\omega t)\) and \(C \sin (\omega t+\phi)\). (a) By expanding \(\sin (\omega t+\phi)\) using the sum of the angles formula, show that the two expressions are equivalent if \(A=C \cos \phi\) and \(B=C \sin \phi .\) (b) Consequently, show that \(A^{2}+B^{2}=C^{2}\) and that \(\phi\) then satisfies the equation \(\tan \phi=\frac{B}{A}\). (c) Generalize your result to state a proposition about \(A_{1} \sin \left(\omega t+\phi_{1}\right)+A_{2} \sin \left(\omega t+\phi_{2}\right)+A_{3} \sin \left(\omega t+\phi_{3}\right)\). (d) Write an essay, in your own words, that expresses the importance of the identity between \(A \sin (\omega t)+B \cos (\omega t)\) and \(C \sin (\omega t+\phi) .\) Be sure to note that \(|C| \geq \max (|A|,|B|)\) and that the identity holds only when you are forming a linear combination (adding and/or subtracting multiples of single powers) of sine and cosine of the same frequency.

Find the equations of the two tangent lines to the circle \(x^{2}+y^{2}=36\) that go through \((12,0)\). Hint: See Problem 72

In Problems 45 -48, find the coordinates of the point of intersection. Then write an equation for the line through that point perpendicular to the line given first. \(5 x-2 y=5\) \(2 x+3 y=9\) \(2 x+3 y=6\)
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