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Chapter 3: Problem 59

Find an equation of the line passing through the given points. (a) Write the equation in standard form. (b) Write the equation in slope-intercept form if possible. $$ (2,5) \text { and }(1,5) $$

Short Answer

Expert verified
The equation in both slope-intercept form and standard form is \( y = 5 \).

Step by step solution

01

Find the slope (m)

Use the slope formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the given points \( (x_1, y_1) = (2, 5) \) and \( (x_2, y_2) = (1, 5) \): \[ m = \frac{5 - 5}{1 - 2} = \frac{0}{-1} = 0 \] The slope (m) is 0.
02

Find the y-intercept (b)

Use the point-slope form of the line equation: \[ y = mx + b \] With m = 0 and substituting one of the points, say \( (2, 5) \): \[ 5 = 0(2) + b \] \[ b = 5 \]
03

Write the equation in slope-intercept form

Now substitute the slope (m) and y-intercept (b) into the slope-intercept form equation: \[ y = mx + b \] With m = 0 and b = 5, the equation is: \[ y = 0x + 5 \] Simplifying this, we get: \[ y = 5 \]
04

Write the equation in standard form

Standard form of a line's equation is \( Ax + By = C \). We have the equation in slope-intercept form: \[ y = 5 \] Rewrite this as: \[ 0x + y = 5 \] Which simplifies to: \[ y = 5 \] This is also the standard form in this case.

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Key Concepts

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

slope calculation
To start finding the equation of a line, we need to determine the slope, commonly represented by the letter \(m\). The slope measures the steepness or direction of the line.
We use the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here's how it works:
  1. Take two points on the line
  2. Label them as \((x_1, y_1)\) and \((x_2, y_2)\)
  3. Plug these coordinates into the formula

In our given problem, the points are \((2, 5)\) and \((1, 5)\). Plugging these into the formula, we get:
\[ m = \frac{5 - 5}{1 - 2} = \frac{0}{-1} = 0 \]So, the slope \(m\) is 0. This means the line is horizontal and has no rise.
slope-intercept form
Once you know the slope, the next step is to express the line in slope-intercept form.
This form of the line equation is written as:
\[ y = mx + b \]
Where:
  • \(m\) is the slope
  • \(b\) is the y-intercept, the point where the line crosses the y-axis

In our example:
  • We already found \(m = 0\)
  • To find \(b\), use the coordinates of one of the points, say \((2, 5)\)

Plugging these into the equation:
\[ 5 = 0 \cdot 2 + b \]
\[ b = 5 \]
Now that we have both \(m\) and \(b\), we can plug these values back into the slope-intercept formula:
\[ y = 0 \cdot x + 5 \]
Which simplifies to:
\[ y = 5 \]
standard form
Finally, let's convert the equation into standard form. The standard form of a line is written as:
\[ Ax + By = C \]
    Where:
  • \(A\), \(B\), and \(C\) are integers
  • \(A\) and \(B\) cannot both be zero
  • \(A\) should be a non-negative integer (usually)
    For our equation in slope-intercept form, which is \( y = 5 \), we can convert it as follows:
    • In slope-intercept form, it's \( y = 5 \)
    • To move to standard form, it can be written as:\[ 0 \cdot x + y = 5 \]
    Simplifying it, we get:
    \[ y = 5 \]
    In this case, the slope-intercept form and the standard form are essentially the same, since the slope is zero. For lines with a non-zero slope, the transformation would involve rearranging terms to fit the \(Ax + By = C\) format.

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    Most popular questions from this chapter

    Find the \(x\) - and \(y\) -intercepts. Then graph each equation. $$ x-3 y=0 $$

    Find an equation of the line passing through the given points. (a) Write the equation in standard form. (b) Write the equation in slope-intercept form if possible. $$ \left(\frac{1}{2},-3\right) \text { and }\left(-\frac{2}{3},-3\right) $$

    Find the midpoint of each segment with the given endpoints. $$ (6.2,5.8) \text { and }(1.4,-0.6) $$

    Three points that lie on the same straight line are said to be collinear. Consider the points \(A(3,1), B(6,2),\) and \(C(9,3) .\) Find the slope of segment \(A B\)

    Find the slope of the line through each pair of points.\(\left(\text {Hint:} \frac{\frac{a}{b}}{\frac{c}{d}}=\frac{a}{b} \div \frac{c}{d}\right)\). $$ \left(-\frac{2}{9}, \frac{5}{18}\right) \text { and }\left(\frac{1}{18},-\frac{5}{9}\right) $$
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