Solve the System of Equations: x + y = 15, 2x + 2y = 12

Linear Systems with Inconsistent Equations

Choose the correct answer for the following exercise:

{x+y=152x+2y=12 \begin{cases} x+y=15 \\ 2x+2y=12\frac{}{} \end{cases}

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Solve the system of equations
00:05 Isolate X
00:11 Express X in terms of Y
00:16 Substitute X to find Y
00:32 Isolate Y
01:00 We got an illogical expression, therefore there is no solution
01:04 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Choose the correct answer for the following exercise:

{x+y=152x+2y=12 \begin{cases} x+y=15 \\ 2x+2y=12\frac{}{} \end{cases}

2

Step-by-step solution

To solve the system of equations, follow the steps below:

  • Simplify the second equation: Start with 2x+2y=12 2x + 2y = 12 . Divide every term by 2 to simplify it to x+y=6 x + y = 6 .
  • Compare the two equations now: x+y=15 x + y = 15 and x+y=6 x + y = 6 .

Consider these equations:
Since both are simplified to the form x+y=constant x + y = \text{constant} , they describe two parallel lines, given that they have the same coefficients of x x and y y but different constants (15 and 6).

Parallel lines never intersect. Thus, there is no solution for this system of equations, as they represent two distinct parallel lines.

Therefore, the correct answer is: No solution.

3

Final Answer

No solution

Key Points to Remember

Essential concepts to master this topic
  • Rule: Simplify equations to compare coefficients and constants
  • Technique: Divide 2x + 2y = 12 by 2 to get x + y = 6
  • Check: If same coefficients but different constants, no solution exists ✓

Common Mistakes

Avoid these frequent errors
  • Trying to solve inconsistent systems algebraically
    Don't attempt substitution or elimination on x + y = 15 and x + y = 6 = impossible values! This wastes time and creates confusion. Always simplify first to identify if the system has parallel lines with no intersection.

Practice Quiz

Test your knowledge with interactive questions

Solve the following equations:

\( \begin{cases} 2x+y=9 \\ x=5 \end{cases} \)

FAQ

Everything you need to know about this question

Why can't I just solve this like any other system?

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You need to identify the type of system first! When simplified equations have the same coefficients but different constants, they represent parallel lines that never meet.

How do I know if equations are parallel?

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After simplifying, if both equations have the form ax+by=c ax + by = c with identical coefficients but different constants, they're parallel lines with no solution.

What's the difference between no solution and infinite solutions?

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No solution: Same coefficients, different constants (parallel lines)
Infinite solutions: Identical equations after simplification (same line)

Could I have made an algebra mistake?

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Always double-check your simplification! Divide 2x+2y=12 2x + 2y = 12 by 2 to get x+y=6 x + y = 6 . Compare with x+y=15 x + y = 15 - clearly different!

Why does this system have no solution graphically?

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Both equations represent parallel lines with the same slope but different y-intercepts. Parallel lines never intersect, so there's no point that satisfies both equations.

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