Two more Algebra questions :

Drawing a blank on this one, not sure where to start.
Find the equation, in standard form, with all integer coefficients, of the line perpendicular to x + 3y = 12 and passing through (9, -5).

I have half a solution to this one:
Solve the system of equations using the substitution method.
If the answer is a unique solution, present it as an ordered pair: (x, y). If not, specify whether the answer is “no solution” or “infinitely many solutions.”
4x + y = 4
2x + 8y = 0

First I did
4x - 4x + y = 4 - 4x
y = 4 - 4x
then
2x + 8(4-4x) = 0
2x + 32 - 32x = 0
2x + 32 - 32x = 0-32
2x - 32x = -32
-30x = -32
-30x/30 = -32/30
x = 32/30
x = 16/15

then
y = 4 - 4x
y = 4 - 4(16/15)
This is where I am stuck...

So i got x = 16/15 an y = ?

To find the equation, in standard form, of the line perpendicular to x + 3y = 12 and passing through (9, -5), we need to determine the slope of the given line and then use the fact that perpendicular lines have negative reciprocal slopes.

Step 1: Convert the given equation to slope-intercept form (y = mx + b).
x + 3y = 12
3y = -x + 12
y = (-1/3)x + 4

Step 2: Determine the slope of the given line.
In the equation y = (-1/3)x + 4, the coefficient of x (-1/3) is the slope.

Step 3: Find the negative reciprocal of the slope.
In this case, the negative reciprocal of (-1/3) is 3/1 or simply 3.

Step 4: Use the point-slope form of a line to find the equation of the line passing through (9, -5) with a slope of 3.
The point-slope form of a line is given by y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is its slope.

Substituting the values x1 = 9, y1 = -5, and m = 3 into the point-slope formula, we get:
y - (-5) = 3(x - 9)
y + 5 = 3x - 27

Step 5: Convert the equation to standard form (Ax + By = C) with all integer coefficients.
To convert the equation y + 5 = 3x - 27 to standard form, we move all the terms to one side:
3x - y = 32

Therefore, the equation, in standard form, with all integer coefficients, of the line perpendicular to x + 3y = 12 and passing through (9, -5) is 3x - y = 32.

Now let's solve the system of equations using the substitution method with the equations 4x + y = 4 and 2x + 8y = 0.

Step 1: Solve one equation for one variable in terms of the other variable.
We already have y = 4 - 4x from the first equation.

Step 2: Substitute the expression for the variable in terms of the other variable into the other equation.
Substituting y = 4 - 4x into the second equation 2x + 8y = 0, we have:
2x + 8(4 - 4x) = 0
2x + 32 - 32x = 0
-30x = -32
x = (-32)/(-30)
x = 16/15

Step 3: Find the value of the other variable.
To find y, we substitute the found value for x into y = 4 - 4x:
y = 4 - 4(16/15)

To simplify this further, we can multiply both numerator and denominator of 16/15 by 15 to get a common denominator:
y = 4 - 4(16/15)
y = 4 - (64/15)

To combine the terms, we need a common denominator:
y = (60/15) - (64/15)
y = (-4/15)

Therefore, the solution to the system of equations is x = 16/15 and y = -4/15.

For the first question, finding the equation of a line perpendicular to a given line and passing through a given point involves a few steps:

1. Find the slope of the given line: To do this, rearrange the given equation x + 3y = 12 into slope-intercept form (y = mx + b), where m represents the slope. In this case, solve for y: 3y = -x + 12, so y = (-1/3)x + 4. Thus, the slope of the given line is -1/3.

2. Determine the slope of the perpendicular line: Rotate the slope of the given line by taking the negative reciprocal. The negative reciprocal of -1/3 is 3/1, which is equal to 3.

3. Use the point-slope form to find the equation: Point-slope form is given by y - y1 = m(x - x1), where (x1, y1) represents the given point and m represents the slope of the line. Plugging in the values (9, -5) for (x1, y1) and 3 for m, we get y - (-5) = 3(x - 9). Simplifying this equation gives y + 5 = 3x - 27.

4. Convert the equation to standard form: Standard form of a linear equation is Ax + By = C, where A, B, and C are integers. To convert the equation y + 5 = 3x - 27 to standard form, rearrange the terms: 3x - y = 32. Multiply the equation by -1 to make all the coefficients integers: -3x + y = -32.

Therefore, the equation in standard form of the line perpendicular to x + 3y = 12 and passing through (9, -5) is -3x + y = -32.

For the second question, you have correctly solved the first part of the system of equations, which is finding x = 16/15. To proceed and find y, substitute this value of x back into one of the original equations, such as y = 4 - 4x.

Substituting x = 16/15 into y = 4 - 4x gives y = 4 - 4(16/15). Simplifying this equation:

y = 4 - (64/15) = (60/15) - (64/15) = -4/15.

Therefore, the solution to the system of equations 4x + y = 4 and 2x + 8y = 0 is (x, y) = (16/15, -4/15).