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Chapter 10: Problem 1

Write an equation of the specified straight line. The line through the point \((1,-2)\) that is parallel to the line with equation \(x+2 y=5\)

Short Answer

Expert verified
The equation is \(y = -\frac{1}{2}x - \frac{3}{2}\).

Step by step solution

01

Determine the Slope of the Given Line

The given line's equation is \(x + 2y = 5\). To find its slope, we need to put it in the slope-intercept form \(y = mx + b\). Start by isolating \(y\):\[2y = -x + 5\] Then divide everything by 2:\[y = -\frac{1}{2}x + \frac{5}{2}\] Thus, the slope \(m\) of the given line is \(-\frac{1}{2}\).
02

Use the Same Slope for the Parallel Line

Lines that are parallel share the same slope. Therefore, the line we need to write also has a slope of \(-\frac{1}{2}\).
03

Find the Equation Using Point-Slope Form

With the point \((1, -2)\) and slope \(m = -\frac{1}{2}\), we use the point-slope form of a line equation, \(y - y_1 = m(x - x_1)\). Substituting the point and slope into the formula gives:\[y + 2 = -\frac{1}{2}(x - 1)\]
04

Simplify to Slope-Intercept Form

Expand and simplify the equation from Step 3:\[y + 2 = -\frac{1}{2}x + \frac{1}{2}\] Subtract 2 from both sides to isolate \(y\):\[y = -\frac{1}{2}x + \frac{1}{2} - 2\] Simplify the constant terms:\[y = -\frac{1}{2}x - \frac{3}{2}\] This is the equation of the line in slope-intercept form.

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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 line's equation is a straightforward way to express a straight line. It is given by \(y = mx + b\), where \(m\) represents the slope, and \(b\) indicates the y-intercept. The slope \(m\) tells us how steep the line is, and it is the number that shows the change in \(y\) for every one unit change in \(x\). The y-intercept \(b\) is the point where the line crosses the y-axis.
  • Easy to identify the slope \(m\) and y-intercept \(b\).
  • Quickly understand how line behaves as \(x\) changes.
In the exercise, once we had the line's equation \(y = -\frac{1}{2}x + \frac{5}{2}\), it directly told us that the slope was \(-\frac{1}{2}\) and provided a simple way to generate a parallel line once we applied the point-slope form with these elements.
Point-Slope Form
The point-slope form of the equation of a line is a handy way to write the equation when you know a point on the line and the slope. It is expressed as \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line, and \(m\) is the slope. This form is beneficial because you can immediately apply it without rearranging the equation to identify the slope or a point.
  • Works directly with known slope and point.
  • Easily rewritten into slope-intercept form.
For example, in the problem, using the point (1, -2) and the slope \(-\frac{1}{2}\), we used the formula to write \(y + 2 = -\frac{1}{2}(x - 1)\). Simplifying gives us the same line in the traditional form: \(y = -\frac{1}{2}x - \frac{3}{2}\).
Parallel Lines
Parallel lines are fascinating because they maintain a consistent distance from one another and never cross. In terms of equations, parallel lines have identical slopes. This characteristic makes determining parallel lines straightforward once the slope is known.
  • Same slope suggests lines are parallel.
  • Parallel lines never intersect.
In the given exercise, we started with the line \(x + 2y = 5\) and found its slope to be \(-\frac{1}{2}\). A line parallel to this one through the point (1, -2) would also have the same slope, \(-\frac{1}{2}\), and thus, by applying the point-slope form, we identified its equation.

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