Please help! I am so lost! The question is "R and S are complementary angles and U and V are also complementary angles. If R=y-2, S=2x+3, U=2x-y and V=x-1, find the values of x, y, R, S, U and V."
R+S = 90
U+V = 90
so,
y-2 + 2x+3 = 90
2x-y + x-1 = 90
2x+y = 89
3x-y = 91
5x = 180
x = 36
and now you can take it from there.
To find the values of x, y, R, S, U, and V in this problem, we can use the information provided about complementary angles.
First, let's review what it means for two angles to be complementary. Complementary angles are two angles whose measures add up to 90 degrees.
Given that R and S are complementary angles, we can set up an equation:
R + S = 90
Substituting the given values for R and S, we have:
(y - 2) + (2x + 3) = 90
Next, we can expand and combine like terms:
y - 2 + 2x + 3 = 90
y + 2x + 1 = 90
We can simplify further by subtracting 1 from both sides:
y + 2x = 89
Now, let's examine the angles U and V, which are also complementary angles. Using the same logic, we can set up an equation:
U + V = 90
Substituting the given values for U and V, we have:
(2x - y) + (x - 1) = 90
Expanding and combining like terms:
3x - y - 1 = 90
Adding 1 to both sides:
3x - y = 91
Now we have a system of equations:
y + 2x = 89 (Equation 1)
3x - y = 91 (Equation 2)
To solve this system, we can use various methods such as substitution or elimination. Let's use the elimination method in this case.
Multiply both sides of Equation 1 by -1:
-y - 2x = -89
Now, add Equation 1 and Equation 2 together:
(-y - 2x) + (3x - y) = -89 + 91
Simplifying further:
-2x - y + 3x - y = 2
Combine like terms:
x - 2y = 2
Now we have a new equation (Equation 3) to work with.
Next, let's solve equations (Equation 2) and (Equation 3) simultaneously.
We can multiply Equation 3 by 3:
3(x - 2y) = 3(2)
Expanding:
3x - 6y = 6
Now, we can add Equation 2 and the new equation together:
3x - y + (3x - 6y) = 91 + 6
Combine like terms:
6x - 7y = 97
We now have a new equation (Equation 4).
To solve equations (Equation 4) and (Equation 1) simultaneously, let's multiply Equation 4 by 2:
2(6x - 7y) = 2(97)
Expanding:
12x - 14y = 194
Now, add Equation 1 and the new equation together:
12x - 14y + (y + 2x) = 194 + 89
Combine like terms:
14x - 13y = 283
This gives us a new equation (Equation 5).
Now, let's solve equations (Equation 4) and (Equation 5) simultaneously.
Let's multiply Equation 4 by 14:
14(6x - 7y) = 14(97)
Expanding:
84x - 98y = 1358
Now, multiply Equation 5 by 12:
12(14x - 13y) = 12(283)
Expanding:
168x - 156y = 3396
Now, add the two new equations together:
84x - 98y + 168x - 156y = 1358 + 3396
Combine like terms:
252x - 254y = 4754
To simplify the equation, let's divide all terms by 2:
126x - 127y = 2377
Now we have a new equation (Equation 6).
We now have a system of two equations:
126x - 127y = 2377 (Equation 6)
14x - 13y = 283 (Equation 5)
To solve this system, we can again use the method of substitution or elimination.
Let's use the substitution method to solve for x.
First, we can isolate x in Equation 5:
14x = 13y + 283
Dividing all terms by 14:
x = (13y + 283)/14
Now, we can substitute this expression for x in Equation 6:
126((13y + 283)/14) - 127y = 2377
Simplifying further:
9y + 1957 - 127y = 2377
Combine like terms:
-118y = 420
Dividing both sides by -118:
y = -3.559
Now that we have the value of y, we can substitute it back into Equation 5 to solve for x:
14x - 13(-3.559) = 283
Simplifying further:
14x + 46.267 = 283
Subtracting 46.267 from both sides:
14x = 236.733
Dividing both sides by 14:
x = 16.9095
Now that we have the values of x and y, we can substitute them back into the respective equations to find the values of R, S, U, and V.
R = y - 2 = -3.559 - 2 = -5.559
S = 2x + 3 = 2(16.9095) + 3 = 37.819
U = 2x - y = 2(16.9095) - (-3.559) = 37.378
V = x - 1 = 16.9095 - 1 = 15.9095
Therefore, the values of x, y, R, S, U, and V are:
x = 16.9095
y = -3.559
R = -5.559
S = 37.819
U = 37.378
V = 15.9095