if a=3x+2y and b=5x-4y, find x and y in terms of a and b
What I did:
2a=6x+4y
+b=5x-4y
---------
2a+b=x
2(3x+2y)+(5x-4y)=x
6x+4y+5x-4y=x
11x=x
11x dvided by x = 11
...that isn't the correct answer. The correct answer (according to text book) is
x=2/11a+1/11b
y=5/22a-3/22b
I'm puzzled on how to do this with all the letters. I also don't know how to find out y. I think I did a bit of figuring for x right, right?
you made an error in the initial addition.
2a = 6x + 4y
b = 5x - 4y
add them:
2a + b = 11x
x = (2a+b)/11 or 2a/11 + b/11 , which is the given answer
now plug the x back into the first equation:
5x - 4y = b
5(2a+b)/11 - 4y = b
times 11
5(2a + b) - 44y = 11b
10a + 5b - 44y = 11b
-44y = -10a + 6b
y = (-10a + 6b)/-44
= (5a - 3b)/22 , which is the same as the given y
Ah, solving for variables can be a bit tricky sometimes. Let's try a different approach and use some clown humor to make it more enjoyable!
To solve for x and y in terms of a and b, we can use a method called "elimination." But, since we're clowns here, let's call it "clownination" instead!
First, let's clownin... err, eliminate y. We'll multiply equation a by 4 and equation b by 2, so the coefficients of y become opposite:
4a = 12x + 8y
2b = 10x - 8y
Next, let's add these clownin... um, equations together to eliminate y:
4a + 2b = 12x + 8y + 10x - 8y
4a + 2b = 22x
Now, let's clownmeh... isolate x:
22x = 4a + 2b
x = (4a + 2b) / 22
x = (2a + b) / 11
Ta-da! We've found x in terms of a and b. Now, let's try to find y using another clownination method.
We'll multiply equation a by 5 and equation b by 3, so the coefficients of x become opposite:
5a = 15x + 10y
3b = 15x - 12y
Now, let's subtract these clownin... I mean, equations to eliminate x:
5a - 3b = 15x + 10y - 15x + 12y
5a - 3b = 22y
Lastly, let's clowngicide... isolate y:
22y = 5a - 3b
y = (5a - 3b) / 22
And there we have it! We've found both x and y in terms of a and b:
x = (2a + b) / 11
y = (5a - 3b) / 22
I hope this clownmusing approach made it easier to understand! Remember to always have a little fun while solving equations, even if it involves clownery!
To solve for x and y in terms of a and b, you need to eliminate one variable at a time. Here's a step-by-step solution:
1. Start with the given equations:
a = 3x + 2y (Equation 1)
b = 5x - 4y (Equation 2)
2. Multiply Equation 1 by 2 and Equation 2 by 3 to make the coefficients of y equal:
2a = 6x + 4y (Equation 3)
3b = 15x - 12y (Equation 4)
3. Add Equation 3 and Equation 4 to eliminate y:
2a + 3b = 6x + 4y + 15x - 12y
2a + 3b = 21x - 8y (Equation 5)
4. Now, let's solve Equation 5 for x:
21x = 2a + 3b + 8y
x = (2a + 3b + 8y) / 21 (Equation 6)
5. Substitute the value of x from Equation 6 into Equation 1:
a = 3((2a + 3b + 8y) / 21) + 2y
a = (6a + 9b + 24y) / 21 + 2y
Multiply both sides by 21 to eliminate the denominator:
21a = 6a + 9b + 24y + 42y
15a = 6a + 9b + 66y
9b + 66y = 15a - 6a
9b + 66y = 9a
Divide both sides by 9:
b + 11y = a (Equation 7)
6. Rearrange Equation 7 to solve for y:
11y = a - b
y = (a - b) / 11 (Equation 8)
7. Substitute the value of y from Equation 8 into Equation 6:
x = (2a + 3b + 8((a - b) / 11)) / 21
x = (2a + 3b + (8a - 8b) / 11) / 21
Multiply both sides by 11 to eliminate the denominator:
11x = 2a + 3b + 8a - 8b
11x = 10a - 5b
Divide both sides by 11:
x = (10a - 5b) / 11 (Equation 9)
Finally, the solutions for x and y in terms of a and b are:
x = (10a - 5b) / 11
y = (a - b) / 11
So, the correct answer is:
x = (10/11)a - (5/11)b
y = (1/11)a - (1/11)b
To find x and y in terms of a and b, we can solve the given equations simultaneously. Here's the correct step-by-step method:
1. Start with the given equations:
a = 3x + 2y
b = 5x - 4y
2. Multiply the first equation by 2 and add it to the second equation:
2a + b = 6x + 4y + 5x - 4y
2a + b = 11x
rearrange this equation to solve for x:
11x = 2a + b
x = (2a + b) / 11
3. Now, substitute the value of x into one of the original equations, let's use the first equation:
a = 3x + 2y
substitute the value of x from step 2:
a = 3((2a + b)/11) + 2y
4. Simplify the equation:
a = (6a + 3b)/11 + 2y
5. Multiply through by 11 to eliminate the fraction:
11a = 6a + 3b + 22y
6. Rearrange the equation to solve for y:
22y = 11a - 6a - 3b
22y = 5a - 3b
y = (5a - 3b) / 22
So, the solutions for x and y in terms of a and b are:
x = (2a + b) / 11
y = (5a - 3b) / 22
Your attempt in finding x was close, but when combining the two equations, you made a mistake. Here's the correct way step-by-step:
2a = 6x + 4y
+b = 5x - 4y
When adding these equations, instead of 2a + b = x, it should be 2a + b = 11x. Then in simplifying further, you made an error when cancelling out the y terms. By following the correct steps, you will obtain the desired solution.