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Magazine Chapter 8: Problem 16
Write an equation of the line passing through the given points. Write the equation in standard form \(A x+B y=C\). See Example 2 . (8,-3) and (4,-8)
Short Answer
Expert verified
The equation of the line is \(5x - 4y = 52\).
Step by step solution
01
Calculate the slope of the line
To find the slope \( m \) of the line through the points \((x_1, y_1) = (8, -3)\) and \((x_2, y_2) = (4, -8)\), use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substitute the values to get \( m = \frac{-8 + 3}{4 - 8} = \frac{-5}{-4} = \frac{5}{4} \). The slope of the line is \(\frac{5}{4}\).
02
Write the point-slope form of the equation
Using the point-slope form of a line's equation, which is \( y - y_1 = m(x - x_1) \), substitute the slope \( m = \frac{5}{4} \) and the point \((8, -3)\) into the equation: \( y + 3 = \frac{5}{4}(x - 8) \).
03
Simplify to slope-intercept form
Expand the equation \( y + 3 = \frac{5}{4}(x - 8) \) to get \( y + 3 = \frac{5}{4}x - 10 \). Then, isolate \( y \) by subtracting 3 from both sides: \( y = \frac{5}{4}x - 13 \).
04
Convert to standard form
To convert \( y = \frac{5}{4}x - 13 \) into standard form \( Ax + By = C \), first eliminate the fraction by multiplying every term by 4: \( 4y = 5x - 52 \). Then rearrange it to \( -5x + 4y = -52 \). Multiply the entire equation by -1 to get integer coefficients with a positive \( x \)-term: \( 5x - 4y = 52 \).
05
Verify the equation with the original points
Substitute the points \((8, -3)\) and \((4, -8)\) into the final equation \( 5x - 4y = 52 \) to verify correctness. For \((8, -3)\): \(5(8) - 4(-3) = 40 + 12 = 52\). For \((4, -8)\): \(5(4) - 4(-8) = 20 + 32 = 52\). Both points satisfy the equation.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Slope Calculation
When we want to express a line mathematically, one of the first things we calculate is its slope. The slope measures how steep a line is, and it is denoted by the letter \( m \). A crucial formula for finding the slope of a line passing through two points, \( (x_1, y_1) \) and \( (x_2, y_2) \), is:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]This formula gives us a ratio representing the vertical change (rise) over the horizontal change (run).
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]This formula gives us a ratio representing the vertical change (rise) over the horizontal change (run).
- Calculate \( y_2 - y_1 \) to find how much the line rises or falls.
- Calculate \( x_2 - x_1 \) to find how far along the x-axis the line moves.
- Divide these two values to get the slope.
Point-Slope Form
Once we have the slope, we can express the line in the point-slope form. This form is particularly useful because it clearly shows the slope and a specific point the line passes through. The point-slope equation is written as:
\[y - y_1 = m(x - x_1)\]Here:
\[y + 3 = \frac{5}{4}(x - 8)\]This equation tells us that any point \( (x, y) \) on the line maintains the slope of \( \frac{5}{4} \) away from the given point \( (8, -3) \).
\[y - y_1 = m(x - x_1)\]Here:
- \( m \) is the slope we calculated.
- \( (x_1, y_1) \) is a point on the line.
\[y + 3 = \frac{5}{4}(x - 8)\]This equation tells us that any point \( (x, y) \) on the line maintains the slope of \( \frac{5}{4} \) away from the given point \( (8, -3) \).
Slope-Intercept Form
The slope-intercept form is a straightforward way to describe lines where you can quickly identify the slope and y-intercept (where the line crosses the y-axis). It's written as:
\[y = mx + b\]Where:
\[y = mx + b\]Where:
- \( m \) is the slope.
- \( b \) is the y-intercept.
- Distribute the slope on the right side: \( y + 3 = \frac{5}{4}x - 10 \).
- Subtract 3 from both sides to solve for \( y \): \( y = \frac{5}{4}x - 13 \).
Standard Form
The standard form of a line's equation presents it in a way that's often preferred when dealing with integer coefficients. It's expressed as:
\[Ax + By = C\]Where \( A \), \( B \), and \( C \) are integers, and \( A \) is usually non-negative. Converting from slope-intercept to standard form requires eliminating any fractions and ensuring integer coefficients:
\[Ax + By = C\]Where \( A \), \( B \), and \( C \) are integers, and \( A \) is usually non-negative. Converting from slope-intercept to standard form requires eliminating any fractions and ensuring integer coefficients:
- Start with the equation in slope-intercept form: \( y = \frac{5}{4}x - 13 \).
- Clear the fractions by multiplying every term by 4: \( 4y = 5x - 52 \).
- Rearrange to standard form: \( -5x + 4y = -52 \).
- To meet convention and have a positive \( A \), multiply by -1: \( 5x - 4y = 52 \).
