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Chapter 10: Problem 9

Find an equation of the line described. Leave the solution in the form \(A x+B y=C\). The line has slope \(m=-\frac{2}{3}\) and contains \((0,5)\)

Short Answer

Expert verified
The equation of the line is \( 2x + 3y = 15 \).

Step by step solution

01

Use Point-Slope Form

To find the equation of a line, start by using the point-slope form of a line: \( y - y_1 = m(x - x_1) \). Here, the slope \( m = -\frac{2}{3} \) and the point \( (x_1, y_1) = (0, 5) \). Substituting these values, the equation becomes \( y - 5 = -\frac{2}{3}(x - 0) \).
02

Simplify the Equation

Simplify the equation from Step 1. Since \( x_1 = 0 \), this becomes \( y - 5 = -\frac{2}{3}x \). Then, add 5 to both sides to isolate \( y \): \( y = -\frac{2}{3}x + 5 \).
03

Convert to Standard Form

To convert the equation from slope-intercept form \( y = mx + b \) to standard form \( Ax + By = C \), first eliminate the fraction by multiplying the entire equation by 3: \( 3y = -2x + 15 \). Next, rearrange the terms to obtain: \( 2x + 3y = 15 \).

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Key Concepts

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

Point-Slope Form
To find the equation of a line, you can start with the point-slope form, which is a very handy tool in coordinate geometry. This form is particularly useful when you have the slope of the line and one point through which the line passes. The point-slope form equation is expressed as:\[y - y_1 = m(x - x_1)\]Where:
  • \(m\) is the slope of the line.
  • \((x_1, y_1)\) is a specific point on the line.
This format is perfect for quickly plugging in the values without needing to alter the form too much. For instance, if you're given a slope of \(-\frac{2}{3}\) and a point \((0, 5)\), you simply substitute these into the formula.You end up with:\[y - 5 = -\frac{2}{3}(x - 0)\]This creates a straightforward starting point for deriving other forms of the linear equation.
Slope-Intercept Form
After finding the equation in point-slope form, it's often useful to convert it into slope-intercept form. This makes identifying the slope and y-intercept much easier at a glance. The slope-intercept form is given by:\[y = mx + b\]Where:
  • \(m\) is the slope of the line.
  • \(b\) is the y-intercept, which is the point where the line crosses the y-axis.
To convert from point-slope to slope-intercept form, we need to simplify the equation. For example, if you have:\[y - 5 = -\frac{2}{3}x\]then by rearranging and solving for \(y\), add 5 to both sides:\[y = -\frac{2}{3}x + 5\]This gives you a clear view of the line's slope of \(-\frac{2}{3}\) and a y-intercept of 5. It's often the most accessible form for graphing.
Standard Form
The standard form of a line gives you another way to express linear equations, especially useful for some algebraic operations and analysis. The standard form is:\[Ax + By = C\]Where:
  • \(A\), \(B\), and \(C\) are integers.
  • \(A\) should be positive.
To convert from slope-intercept to standard form, you often need to eliminate fractions and rearrange the terms. For a line equation like \(y = -\frac{2}{3}x + 5\), start by getting rid of the fraction by multiplying every term by 3:\[3y = -2x + 15\]Next, rearrange the terms to get:\[2x + 3y = 15\]This version is particularly useful for evaluating intersections and solutions in systems of equations due to its symmetry and balance.

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Most popular questions from this chapter

Where \(a\) and \(b\) are real numbers, \((a+b, a-2 b)\) is the point of intersection for the lines \(3 x-2 y=19\) and \(2 x-5 y=9 .\) Find the values of \(a\) and \(b\).

The line \((x, y, z)=(0,0,0)+n(1,1,2)\) intersects the sphere \(x^{2}+y^{2}+z^{2}=54\) in two points. Find each point.

Draw an ideally placed figure in the coordinate system; then name the coordinates of each vertex of the figure. a) A trapezoid b) A trapezoid (midpoints of sides are needed)

Find \(a\) such that the points \(A(1,3), B(4,5),\) and \(C(a, a)\) are collinear.

Apply the Midpoint Formula. A circle has its center at the point \((-2,3) .\) If one endpoint of a diameter is at \((3,-5),\) find the other endpoint of the diameter.
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