Question

Given the line \(5 x-4 y=9\), give the equation of a line parallel and passing through \((9,5)\). A. \(4 x-5 y=9\) B. \(9 x-5 y=4\) C. \(5 x-4 y=25\) D. \(5 x+4 y=9\)

Short answer

Option C: \(5x - 4y = 25\).

Step-by-step solution

- Identify the slope of the given line

Given the equation of the line is: \[5x - 4y = 9\]Rewrite it in the slope-intercept form \[y = mx + b\]First, solve for y:\[-4y = -5x + 9\]Divide by -4:\[ y = \frac{5}{4}x - \frac{9}{4} \]The slope (m) of the given line is \(\frac{5}{4}\).

- Use the slope for the parallel line

A line parallel to the given line will have the same slope.So, the slope of the new line that is parallel to the given line is also \(\frac{5}{4}\).

- Use point-slope form to find the equation

The point-slope form of a line is given by:\[y - y_1 = m(x - x_1)\]Here, \((x_1, y_1) = (9, 5)\) and \(m = \frac{5}{4}\).Substitute these values into the point-slope form:\[y - 5 = \frac{5}{4}(x - 9)\]Simplify the equation to get it into standard form.

- Simplify to standard form

First, distribute \(\frac{5}{4}\) on the right side:\[ y - 5 = \frac{5}{4}x - \frac{5 \cdot 9}{4}\]\[ y - 5 = \frac{5}{4}x - \frac{45}{4}\]Multiply every term by 4 to clear the fraction:\[4y - 20 = 5x - 45\]Rearrange to standard form:\[5x - 4y = 25\]

- Compare with given options

Among the given options, the one that matches our derived equation is option C: \[5x - 4y = 25\].

Key concepts

Slope-Intercept Form of a Line

The slope-intercept form of a line is one of the most common ways to represent a linear equation. It is written as:
y = mx + b
Here, 'y' is the dependent variable, 'x' is the independent variable, 'm' represents the slope of the line, and 'b' is the y-intercept (the point where the line crosses the y-axis). Understanding the slope-intercept form is essential when dealing with linear equations because it gives a direct view of the slope and the starting point of the line.

Key Points:

• The slope (m) indicates the steepness of the line; a positive slope means the line ascends from left to right, and a negative slope means it descends.
• The y-intercept (b) is the value of y when x equals zero. It's where the line intersects the y-axis.

In the given exercise, the original equation 5x - 4y = 9 was converted to slope-intercept form to find the slope:

• First step: Rewrite the equation to isolate y: -4y = -5x + 9
• Second step: Divide by -4: y = (5/4)x - 9/4

This reveals the slope (m) as 5/4.

Using Point-Slope Form

The point-slope form is another useful way to express the equation of a line, especially when you have a point on the line and the slope. It is given by:
y - y_1 = m(x - x_1)
Here, (x1, y1) is a known point on the line, and 'm' is again the slope. Using this form simplifies finding the equation of a line passing through a specific point with a known slope.

Let's apply this to our problem: We need a parallel line passing through (9, 5) with the same slope (5/4).
So we substitute (x1, y1) = (9, 5) and m = 5/4 into the point-slope form: y - 5 = 5/4(x - 9)

• First step: Distribute 5/4 on the right side: y - 5 = 5/4x - 45/4
• Second step: Simplify and rearrange terms if needed. For instance, to change it to standard form.

This makes it easy to move forward to the next form if necessary.

Standard Form of a Line

The standard form of a linear equation is another common way to express a line. It is written as:
Ax + By = C
Here, A, B, and C are integers, and A should be positive. The standard form is helpful since it simplifies finding intercepts and can be used readily for certain algebraic tasks.
When converting from point-slope form to standard form, follow these steps:

• First: Multiply every term by a common factor to clear any fractions.
• Second: Move all terms with variables to one side and constants to the other.

For our example, we started from y - 5 = 5/4(x - 9) and distributed the 5/4: y - 5 = 5/4x - 45/4

• Multiplied every term by 4: 4(y - 5) = 5x - 45
• Got rid of parentheses: 4y - 20 = 5x - 45
• Moved all terms to one side: 5x - 4y = 25

The correct answer matched this format, which was Option C: 5x - 4y = 25