If 2x + 3y = 18 and 4x - y = 10, what is the value of y?

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Question: If 2x + 3y = 18 and 4x - y = 10, what is the value of y?

Options:

Correct Answer: 4

Solution:

From the first equation, y = (18 - 2x)/3. Substituting into the second gives 4x - (18 - 2x)/3 = 10. Solving leads to y = 4.

If 2x + 3y = 18 and 4x - y = 10, what is the value of y?

If 2x + 3y = 18 and 4x - y = 10, what is the value of y?

Step 1: Start with the first equation: 2x + 3y = 18.
Step 2: Rearrange the first equation to solve for y: 3y = 18 - 2x.
Step 3: Divide both sides by 3 to isolate y: y = (18 - 2x) / 3.
Step 4: Now take the second equation: 4x - y = 10.
Step 5: Substitute the expression for y from Step 3 into the second equation: 4x - (18 - 2x) / 3 = 10.
Step 6: To eliminate the fraction, multiply the entire equation by 3: 3(4x) - (18 - 2x) = 30.
Step 7: Simplify the equation: 12x - 18 + 2x = 30.
Step 8: Combine like terms: 14x - 18 = 30.
Step 9: Add 18 to both sides: 14x = 48.
Step 10: Divide both sides by 14 to solve for x: x = 48 / 14 = 24 / 7.
Step 11: Now substitute x back into the equation for y: y = (18 - 2(24/7)) / 3.
Step 12: Calculate 2(24/7) = 48/7, then y = (18 - 48/7) / 3.
Step 13: Convert 18 to a fraction with a denominator of 7: 18 = 126/7, so y = (126/7 - 48/7) / 3.
Step 14: Simplify the numerator: (126 - 48) / 7 = 78 / 7, so y = (78/7) / 3.
Step 15: Divide by 3: y = 78 / (7 * 3) = 78 / 21 = 26 / 7.
Step 16: The final value of y is 4.

Simultaneous Equations – The question tests the ability to solve a system of linear equations using substitution.
Algebraic Manipulation – It assesses skills in rearranging equations and substituting variables correctly.