Two more Algebra questions : Having some difficulty with these.
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 = ?
(I did not check you "x" if it is correct or not)
First, you need to simplify 4{16/15)
Then, find a common denominator
New Equation:
y = 60/15 - 64/15
Answer:
y = 4/15
For the first question, to find the equation of a line perpendicular to another line, you need to do the following steps:
1. Find the slope of the given line by rearranging the equation into slope-intercept form (y = mx + b), which means solving for y in terms of x.
For the line given, x + 3y = 12, rearrange it to y = (-1/3)x + 4/3.
This gives you the slope of the given line, which is -1/3.
2. The slope of a line perpendicular to another line is the negative reciprocal of the slope of the given line. So, the perpendicular slope would be 3/1 or just 3.
3. Now that you have the slope of the perpendicular line, you can use the point-slope form of a linear equation to find the equation of the line.
The point-slope form is y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope.
In this case, the point you are given is (9, -5), and the slope is 3.
Plugging these values into the point-slope form, you get:
y - (-5) = 3(x - 9)
4. Simplify the equation:
y + 5 = 3x - 27
5. Rewrite the equation in standard form (ax + by = c) with integer coefficients by moving all the terms to one side:
3x - y = 32
Therefore, the equation of the line that is perpendicular to x + 3y = 12 and passes through the point (9, -5) is 3x - y = 32.
Now, for the second question, let's continue from where you left off:
You correctly found that x = 16/15.
To find y, substitute the value of x back into one of the original equations:
4x + y = 4
Replace x with 16/15:
4(16/15) + y = 4
Multiply through by 15 to remove the fraction:
64 + 15y = 60
Subtract 64 from both sides:
15y = 60 - 64
15y = -4
Divide both sides by 15:
y = -4/15
So, the solution to the system of equations is x = 16/15 and y = -4/15. Thus, the ordered pair is (16/15, -4/15).