Question:
The number of solutions for the system of equations \( 2x + y = 4 \), \( 3x + 2y = 2 \), and \( x + y = -2 \) is:
The number of solutions for the system of equations \( 2x + y = 4 \), \( 3x + 2y = 2 \), and \( x + y = -2 \) is:
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Solve two equations first and verify the obtained point in the remaining equation.
Updated On: May 6, 2026
- \( 1 \)
- \( 2 \)
- \( 3 \)
- infinitely many
- \( 0 \)
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The Correct Option is A
Solution and Explanation
Concept:
A system of linear equations has a solution only if one point satisfies all equations simultaneously.
Step 1: Solve the first two equations.
\[ 2x + y = 4 \] \[ 3x + 2y = 2 \] From first equation: \[ y = 4 - 2x \] Substitute into second equation: \[ 3x + 2(4 - 2x) = 2 \] \[ 3x + 8 - 4x = 2 \] \[ -x = -6 \Rightarrow x = 6 \] Now calculate $y$: \[ y = 4 - 2(6) = -8 \] Thus, the intersection point is: \[ (6,-8) \]
Step 2: Check the point in the third equation.
Third equation: \[ x + y = -2 \] Substitute: \[ 6 + (-8) = -2 \] \[ -2 = -2 \] Hence, the point satisfies the third equation also. Conclusion:
All three equations intersect at one common point. \[ \boxed{\text{One solution}} \]
A system of linear equations has a solution only if one point satisfies all equations simultaneously.
Step 1: Solve the first two equations.
\[ 2x + y = 4 \] \[ 3x + 2y = 2 \] From first equation: \[ y = 4 - 2x \] Substitute into second equation: \[ 3x + 2(4 - 2x) = 2 \] \[ 3x + 8 - 4x = 2 \] \[ -x = -6 \Rightarrow x = 6 \] Now calculate $y$: \[ y = 4 - 2(6) = -8 \] Thus, the intersection point is: \[ (6,-8) \]
Step 2: Check the point in the third equation.
Third equation: \[ x + y = -2 \] Substitute: \[ 6 + (-8) = -2 \] \[ -2 = -2 \] Hence, the point satisfies the third equation also. Conclusion:
All three equations intersect at one common point. \[ \boxed{\text{One solution}} \]
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