Given the equations: x + 2y = 10 and 3x - y = 5, what is the value of x?

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Question: Given the equations: x + 2y = 10 and 3x - y = 5, what is the value of x?

Options:

Correct Answer: 4

Solution:

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

Given the equations: x + 2y = 10 and 3x - y = 5, what is the value of x?

Given the equations: x + 2y = 10 and 3x - y = 5, what is the value of x?

Step 1: Start with the first equation: x + 2y = 10.
Step 2: Rearrange the first equation to solve for x: x = 10 - 2y.
Step 3: Now take the second equation: 3x - y = 5.
Step 4: Substitute the expression for x from Step 2 into the second equation: 3(10 - 2y) - y = 5.
Step 5: Distribute 3 in the equation: 30 - 6y - y = 5.
Step 6: Combine like terms: 30 - 7y = 5.
Step 7: Subtract 30 from both sides: -7y = 5 - 30.
Step 8: Simplify the right side: -7y = -25.
Step 9: Divide both sides by -7 to solve for y: y = 25/7.
Step 10: Now substitute y back into the expression for x: x = 10 - 2(25/7).
Step 11: Calculate 2(25/7): 50/7.
Step 12: Now calculate x: x = 10 - 50/7.
Step 13: Convert 10 to a fraction with a denominator of 7: 10 = 70/7.
Step 14: Now subtract: x = 70/7 - 50/7 = 20/7.
Step 15: Therefore, the value of x is 20/7.

Systems of 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.