If 5x + 3y = 15 and 2x + 4y = 10, what is the value of y?

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Question: If 5x + 3y = 15 and 2x + 4y = 10, what is the value of y?

Options:

Correct Answer: 2

Solution:

From the first equation, 3y = 15 - 5x, so y = (15 - 5x)/3. Substituting into the second gives 2x + 4(15 - 5x)/3 = 10. Solving leads to y = 2.

If 5x + 3y = 15 and 2x + 4y = 10, what is the value of y?

If 5x + 3y = 15 and 2x + 4y = 10, what is the value of y?

Step 1: Start with the first equation: 5x + 3y = 15.
Step 2: Rearrange the first equation to isolate 3y: 3y = 15 - 5x.
Step 3: Divide both sides of the equation by 3 to solve for y: y = (15 - 5x) / 3.
Step 4: Now take the second equation: 2x + 4y = 10.
Step 5: Substitute the expression for y from Step 3 into the second equation: 2x + 4((15 - 5x) / 3) = 10.
Step 6: Simplify the equation: 2x + (60 - 20x) / 3 = 10.
Step 7: Multiply the entire equation by 3 to eliminate the fraction: 3(2x) + (60 - 20x) = 30.
Step 8: This simplifies to 6x + 60 - 20x = 30.
Step 9: Combine like terms: -14x + 60 = 30.
Step 10: Subtract 60 from both sides: -14x = 30 - 60, which simplifies to -14x = -30.
Step 11: Divide both sides by -14 to solve for x: x = -30 / -14, which simplifies to x = 15/7.
Step 12: Now substitute x back into the equation for y: y = (15 - 5(15/7)) / 3.
Step 13: Calculate 5(15/7) = 75/7, so y = (15 - 75/7) / 3.
Step 14: Convert 15 to a fraction with a denominator of 7: 15 = 105/7, so y = (105/7 - 75/7) / 3.
Step 15: Simplify the numerator: y = (30/7) / 3.
Step 16: Divide by 3: y = (30/7) * (1/3) = 30/21, which simplifies to y = 10/7.
Step 17: However, we need to check if y = 2 is a solution by substituting back into the original equations.

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