If x and y satisfy the equations |x| + x + y = 15 and x + |y| − y = 20, then (x − y) equals
Question:
If x and y satisfy the equations |x| + x + y = 15 and x + |y| − y = 20, then (x − y) equals
If x and y satisfy the equations |x| + x + y = 15 and x + |y| − y = 20, then (x − y) equals
Options
Answer: 15
Explanation:
Analyze signs: If x < 0 → |x| + x = 0 → y = 15 Then x + |y| − y = x + 15 − 15 = x = 20 ❌ Not possible So x > 0 → |x| + x = 2x If y > 0 → x + |y| − y = x + y − y = x = 20 → first eq: 2x + y = 15 → 2×20 + y = 15 → y = −25 ❌ Not possible So y < 0 → |y| − y = −y − y = −2y → x − 2y = 20 Solve system: 2x + y = 15 x − 2y = 20 Multiply second eq by 1, first eq by 1: 2x + y = 15 x − 2y = 20 Multiply second eq by 2: 2x − 4y = 40 Subtract first eq: (2x − 4y) − (2x + y) = 40 − 15 → −5y = 25 → y = −5 Then x = 20 − 2(−5) = 20 + 10 = 30 ❌ Wait recalc: Use x − 2y = 20 → x − 2(−5) = x + 10 = 20 → x = 10 ✅ Check 2x + y = 2×10 + (−5) = 20 − 5 = 15 ✅ Compute x − y = 10 − (−5) = 15
Explanation:
Analyze signs: If x < 0 → |x| + x = 0 → y = 15 Then x + |y| − y = x + 15 − 15 = x = 20 ❌ Not possible So x > 0 → |x| + x = 2x If y > 0 → x + |y| − y = x + y − y = x = 20 → first eq: 2x + y = 15 → 2×20 + y = 15 → y = −25 ❌ Not possible So y < 0 → |y| − y = −y − y = −2y → x − 2y = 20 Solve system: 2x + y = 15 x − 2y = 20 Multiply second eq by 1, first eq by 1: 2x + y = 15 x − 2y = 20 Multiply second eq by 2: 2x − 4y = 40 Subtract first eq: (2x − 4y) − (2x + y) = 40 − 15 → −5y = 25 → y = −5 Then x = 20 − 2(−5) = 20 + 10 = 30 ❌ Wait recalc: Use x − 2y = 20 → x − 2(−5) = x + 10 = 20 → x = 10 ✅ Check 2x + y = 2×10 + (−5) = 20 − 5 = 15 ✅ Compute x − y = 10 − (−5) = 15
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