If 9x + 4y = 36 and 2x + 3y = 12, what is the value of y?

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Question: If 9x + 4y = 36 and 2x + 3y = 12, what is the value of y?

Options:

Correct Answer: 3

Solution:

From the second equation, 3y = 12 - 2x. Substituting into the first gives 9x + 4(12 - 2x)/3 = 36. Solving gives y = 3.

If 9x + 4y = 36 and 2x + 3y = 12, what is the value of y?

If 9x + 4y = 36 and 2x + 3y = 12, what is the value of y?

Step 1: Start with the two equations: 9x + 4y = 36 and 2x + 3y = 12.
Step 2: Take the second equation, 2x + 3y = 12, and solve for 3y. To do this, subtract 2x from both sides: 3y = 12 - 2x.
Step 3: Now we have an expression for 3y. Next, we need to express y. Divide both sides of the equation by 3: y = (12 - 2x) / 3.
Step 4: Substitute this expression for y back into the first equation (9x + 4y = 36). Replace y with (12 - 2x) / 3: 9x + 4((12 - 2x) / 3) = 36.
Step 5: Simplify the equation. First, multiply 4 by (12 - 2x): 9x + (48 - 8x) / 3 = 36.
Step 6: To eliminate the fraction, multiply the entire equation by 3: 3(9x) + (48 - 8x) = 3(36).
Step 7: This simplifies to 27x + 48 - 8x = 108.
Step 8: Combine like terms: (27x - 8x) + 48 = 108, which gives 19x + 48 = 108.
Step 9: Subtract 48 from both sides: 19x = 108 - 48, which simplifies to 19x = 60.
Step 10: Divide both sides by 19 to find x: x = 60 / 19.
Step 11: Now, substitute x back into the equation for y: y = (12 - 2(60 / 19)) / 3.
Step 12: Calculate the value of y: y = (12 - 120 / 19) / 3.
Step 13: Convert 12 into a fraction with a denominator of 19: y = ((228 - 120) / 19) / 3.
Step 14: Simplify the numerator: y = (108 / 19) / 3.
Step 15: Divide by 3: y = 108 / (19 * 3) = 108 / 57.
Step 16: Simplify the fraction: y = 3.

Systems of Equations – The question tests the ability to solve a system of linear equations using substitution or elimination methods.
Algebraic Manipulation – The question requires skills in rearranging equations and substituting values correctly.