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Magazine Chapter 0: Problem 33
Find an equation for each line. Then write your answer in the form \(A x+B y+C=0 .\) Through (2,3) and (4,8)
Short Answer
Expert verified
The equation of the line is: \(5x - 2y - 4 = 0\).
Step by step solution
01
Determine the slope of the line
To find the slope of the line through two points \((x_1, y_1) = (2, 3)\) and \((x_2, y_2) = (4, 8)\), use the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Substitute the given points into the formula: \[ m = \frac{8 - 3}{4 - 2} = \frac{5}{2} \]Thus, the slope of the line is \(\frac{5}{2}\).
02
Write the point-slope form of the line
Using the point-slope form of the line equation \[ y - y_1 = m(x - x_1) \]and the point \((x_1, y_1) = (2, 3)\)\ and the slope \(m = \frac{5}{2}\), substitute these values into the formula:\[ y - 3 = \frac{5}{2}(x - 2) \]
03
Simplify to slope-intercept form
Distribute \(\frac{5}{2}\) on the right side of the equation:\[ y - 3 = \frac{5}{2}x - 5 \]Add 3 to both sides to isolate \(y\):\[ y = \frac{5}{2}x - 2 \]
04
Convert to standard form
Multiply every term by 2 to eliminate the fraction:\[ 2y = 5x - 4 \]Rearrange to standard form \(Ax + By + C = 0\) by moving all terms to one side:\[ -5x + 2y + 4 = 0 \]This simplifies to:\[ 5x - 2y = -4 \]
05
Final equation in standard form
Ensure the leading coefficient of \(x\) is positive and confirm the form:\[ 5x - 2y - 4 = 0 \]Thus, the line equation in standard form is \(5x - 2y - 4 = 0\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Slope
The slope of a line is a measure of its steepness and direction. To find the slope between two points, we use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Here, \( m \) represents the slope.
This formula helps us understand how much \( y \) changes for a change in \( x \). It can be positive, negative, zero, or undefined.
In our example, the slope is \( \frac{5}{2} \). This tells us that for every 2 units we move horizontally, the line rises by 5 units.
- The difference in the \( y \)-coordinates is the numerator.
- The difference in the \( x \)-coordinates is the denominator.
This formula helps us understand how much \( y \) changes for a change in \( x \). It can be positive, negative, zero, or undefined.
- A positive slope means the line goes upwards as you move from left to right.
- A negative slope means it goes downwards.
- Zero slope means the line is horizontal.
- An undefined slope means the line is vertical.
In our example, the slope is \( \frac{5}{2} \). This tells us that for every 2 units we move horizontally, the line rises by 5 units.
Point-Slope Form
The point-slope form is useful for writing equations when you know a point on the line and its slope. The form of the equation is:\[ y - y_1 = m(x - x_1) \]
Using this form helps you quickly plug in the values you have to get the equation of a line. This is very handy for calculations and to understand the line's behavior at the point you know.
In our exercise, with point \((2, 3)\) and slope \(\frac{5}{2}\), the equation becomes: \[ y - 3 = \frac{5}{2}(x - 2) \]This form clearly shows how the line passes through a specific point while maintaining the defined slope.
- \( (x_1, y_1) \) is a known point on the line.
- \( m \) is the slope.
Using this form helps you quickly plug in the values you have to get the equation of a line. This is very handy for calculations and to understand the line's behavior at the point you know.
In our exercise, with point \((2, 3)\) and slope \(\frac{5}{2}\), the equation becomes: \[ y - 3 = \frac{5}{2}(x - 2) \]This form clearly shows how the line passes through a specific point while maintaining the defined slope.
Standard Form
The standard form of a line is a widely used format, especially in algebra problems. The general formula is:\[ Ax + By + C = 0 \]
One thing to remember is that this format doesn't directly show the slope or y-intercept. It's often used because it helps in comparing two lines or in finding intersections.
In our example, after manipulations, we ended up with: \[ 5x - 2y - 4 = 0 \]Rewriting the equation from any form to standard form often involves simple algebraic steps like eliminating fractions and rearranging terms.
- \( A, B, \) and \( C \) are constants.
- \( A \) should be a positive integer.
One thing to remember is that this format doesn't directly show the slope or y-intercept. It's often used because it helps in comparing two lines or in finding intersections.
In our example, after manipulations, we ended up with: \[ 5x - 2y - 4 = 0 \]Rewriting the equation from any form to standard form often involves simple algebraic steps like eliminating fractions and rearranging terms.
Slope-Intercept Form
Many students prefer the slope-intercept form because it clearly tells you the slope and the y-intercept of the line. The form is:\[ y = mx + b \]
It's straightforward to see how the line behaves and where it begins on the graph.
In our work, we derived this form from the point-slope form:\[ y = \frac{5}{2}x - 2 \]Here, you can instantly identify that the line's slope is \(\frac{5}{2}\) and it crosses the y-axis at \(-2\). This form is particularly beneficial when plotting or quickly assessing the line's characteristics.
- \( m \) is the slope of the line.
- \( b \) is the y-intercept, meaning where the line crosses the y-axis.
It's straightforward to see how the line behaves and where it begins on the graph.
In our work, we derived this form from the point-slope form:\[ y = \frac{5}{2}x - 2 \]Here, you can instantly identify that the line's slope is \(\frac{5}{2}\) and it crosses the y-axis at \(-2\). This form is particularly beneficial when plotting or quickly assessing the line's characteristics.
