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Chapter 2: Problem 18

Find an equation of the line that satisfies the given conditions. Through \((-3,-5) ; \quad\) slope \(-\frac{7}{2}\)

Short Answer

Expert verified
The equation is \( y = -\frac{7}{2}x - \frac{31}{2} \).

Step by step solution

01

Understand the Formula

The equation of a line in slope-intercept form is given by \( y = mx + b \), where \( m \) represents the slope and \( b \) the y-intercept. We are provided with a point and a slope, and we will utilize these to find the equation.
02

Use Point-Slope Form

The point-slope form of the equation of a line is \( y - y_1 = m(x - x_1) \). Given point \((-3, -5)\) and slope \( m = -\frac{7}{2} \), substitute these values into the formula: \( y - (-5) = -\frac{7}{2}(x - (-3)) \). This simplifies to \( y + 5 = -\frac{7}{2}(x + 3) \).
03

Simplify the Equation

Distribute \(-\frac{7}{2}\) through the terms in the parentheses: \( y + 5 = -\frac{7}{2}x - \frac{21}{2} \).
04

Solve for y

To convert to slope-intercept form, solve for \( y \) by subtracting 5 from both sides: \( y = -\frac{7}{2}x - \frac{21}{2} - 5 \).
05

Simplify Further

Combine the constants on the right side: \( -\frac{21}{2} - 5 \) can be rewritten as \( -\frac{21}{2} - \frac{10}{2} = -\frac{31}{2} \). This provides us the final equation, \( y = -\frac{7}{2}x - \frac{31}{2} \).

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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 of a linear equation is a widely used representation because it easily shows the slope and y-intercept of a line. This form is set up as \( y = mx + b \), where:
  • \( m \) is the slope of the line, indicating its steepness and direction.
  • \( b \) is the y-intercept, the point where the line crosses the y-axis.
The beauty of this form is its simplicity and intuitive grasp. By knowing \( m \) and \( b \), one can quickly visualize the line's behavior on a graph.
To transform an equation from point-slope to slope-intercept form, we solve for \( y \). After inserting the given slope and calculating the intercept, a clear linear equation emerges, ready for graphing or further analysis.
Point-Slope Form
When you're given a point and a slope, the point-slope form provides a straightforward way to construct the equation of a line. It is expressed as \( y - y_1 = m(x - x_1) \), where:
  • \((x_1, y_1)\) are the coordinates of a specific point the line passes through.
  • \( m \) is, as always, the slope.
This form is particularly handy when dealing with problems where a specific point on the line is known. It ties the slope to actual coordinates, making it easier to visualize how the line angles through a particular point. Once the equation is set, further simplifications can convert it into slope-intercept form, which is often more convenient for plotting or solving additional problems.
Linear Equations
Linear equations describe a straight line on a plane and are foundational in algebra. They typically combine constants and variables, forming equations that represent a direct relationship between two quantities.
In their basic forms, such as the slope-intercept (\( y = mx + b \)) and point-slope (\( y - y_1 = m(x - x_1) \)), linear equations provide vital information about the geometric entities they describe.
  • They show the relationship between two variables, often depicting real-world problems like speed versus time or cost versus quantity.
  • Understanding how to manipulate these equations, whether by changing forms or solving for specific variables, is crucial for applying algebra effectively.
By learning to interpret and work with linear equations, one gains a powerful toolset for tackling a wide array of mathematical challenges.

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Most popular questions from this chapter

A small-appliance manufacturer finds that if he produces \(x\) toaster ovens in a month his production cost is given by the equation $$y=6 x+3000$$ (where \(y\) is measured in dollars). (a) Sketch a graph of this linear equation. (b) What do the slope and \(y\) -intercept of the graph represent?

Frequency of Vibration The frequency \(f\) of vibration of a violin string is inversely proportional to its length \(L\) . The constant of proportionality \(k\) is positive and depends on the tension and density of the string. (a) Write an equation that represents this variation. (b) What effect does doubling the length of the string have on the frequency of its vibration?

Find an equation of the line that satisfies the given conditions. Through \((-1,2) ;\) parallel to the line \(x=5\)

Find the slope and \(y\)-intercept of the line and draw its graph. \(3 x+4 y-1=0\)

Find an equation of the perpendicular bisector of the line segment joining the points \(A(1,4)\) and \(B(7,-2) .\)
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