Question

Write an equation of the line containing (8,-6) and parallel to the graph of 3x-4y=9

Answer 1

y= 3/4x+12

Explanation:

We can find the equation of the line by using y= mx+c where m is the gradient (slope) and c is the y-intercept. The y-intercept is the y coordinate when x is 0. The equation of the other line is not in the form of y=mx+c so we have to rearrange the equation.

3x-4y=9

3x=9+4y

4y=3x-9

y= (3x-9)/4

We need it in the form of y= mx+c so y=
`(3x)/(4)` -
`(9)/(4)`

That means that m= 3/4

Now we can use y-y1= m (x-x1)

Plug in the values. y-(-6)= 3/4(x-8)

y+6= 3/4x-(3/4)×-8

y+6= 3/4x+6

y= 3/4x+6+6

y= 3/4x+12

Answer 2

3x - 4y = 48 (standard form) or y = (3/4)x - 12 (slope-intercept form)

Explanation:

We are given the equation of a line: 3x - 4y = 9, and we want to find the equation of a line that is parallel to this line and passes through the point (8, -6).

Step 1: Find the slope of the given line.

To do this, we rearrange the given equation into the slope-intercept form (y = mx + b), where "m" is the slope:

⇒ 3x - 4y = 9

⇒ -4y = -3x + 9

∴ y = (3/4)x - 9/4

The slope of the given line is 3/4.

Step 2: Write the equation of the parallel line using the point-slope form (y - y₁ = m(x - x₁)).

Since the new line is parallel, it will have the same slope of 3/4. We can use the point-slope form of the equation to write the line using the given point (8, -6):

⇒ y - (-6) = (3/4)(x - 8)

Step 3: Simplify the equation.

⇒ y + 6 = (3/4)(x - 8)

Step 4: Convert the equation to standard form (Ax + By = C) or slope-intercept form (y = mx + b).

Putting the equation in standard form:

To do this, we eliminate the fraction by multiplying both sides by 4:

⇒ 4(y + 6) = 3(x - 8)

⇒ 4y + 24 = 3x - 24

Now, rearrange the equation:

∴ 3x - 4y = 48

Putting the equation in slope-intercept form:

To do this, we eliminate the fraction by multiplying both sides by 4:

⇒ 4(y + 6) = 3(x - 8)

⇒ 4y + 24 = 3x - 24

Now, rearrange the equation:

⇒ 4y = 3x - 48

∴ y = (3/4)x - 12

So, the equation of the line parallel to the graph of 3x - 4y = 9 and passing through the point (8, -6) is 3x - 4y = 48 or y = (3/4)x - 12.

`\hrulefill`

Additional Knowledge:

Equation of a line: An equation that represents a straight line on a coordinate plane. The general form of the equation of a line is given by y = mx + b, where "m" is the slope of the line, and "b" is the y-intercept (the point where the line crosses the y-axis).

Slope: The slope (a.k.a rate of change, gradient, pitch) of a line is a measure of how steep or slanted the line is. It is represented by the letter "m" in the slope-intercept form of the equation as well as the point-slope form of the equation. The slope measures the change in "y" (vertical change) divided by the change in "x" (horizontal change) between any two points on the line.

Slope-intercept form: A way of writing the equation of a line in the form y = mx + b, where "m" represents the slope of the line, and "b" represents the y-intercept, which is the point where the line intersects the y-axis.

Parallel lines: Two lines are parallel if they never intersect, and they have the same slope. In other words, if two lines have the same slope, they are parallel.

Point-slope form: Another way of writing the equation of a line. It is written as y - y₁ = m(x - x₁), where (x₁, y₁) are the coordinates of a point on the line, and "m" is the slope of the line.