Solve the Linear System: Equations x+y=0 and x+y=10

Q: How can I quickly spot an inconsistent system?

Look for equations with identical left sides but different right sides. Like $ x+y=0 $ and $ x+y=10 $ - same expression can't equal two different numbers!

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

No solution: Equations contradict (like $ x+y=0 $ and $ x+y=10 $). Infinite solutions: Equations are identical (like $ x+y=0 $ and $ 2x+2y=0 $).

Q: Should I graph this to verify there's no solution?

Great idea! Both equations represent parallel lines with the same slope but different y-intercepts. Parallel lines never intersect, confirming no solution exists.

Q: What if I accidentally try to solve it anyway?

You'll get a false statement like $ 0 = 10 $. This contradiction confirms the system has no solution - it's actually a helpful verification method!

Q: Are inconsistent systems common in real life?

Yes! Think of impossible situations: 'I need 5 apples and 5 apples costs $10' versus '5 apples costs $20'. These contradictory conditions have no solution.

Linear Systems with Inconsistent Equations

$x+y=0$

$x+y=10$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:05 Subtract between the equations

00:16 Group terms

00:23 We got an illogical expression, therefore there is no solution

00:30 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$x+y=0$

$x+y=10$

Step-by-step solution

To solve this system of equations, let's analyze the given equations:

The first equation is $x + y = 0$ .
The second equation is $x + y = 10$ .

Notice that the left-hand side of both equations is the same, $x + y$ , but the right-hand sides are different: 0 and 10, respectively.

This means that there is no possible way for $x + y$ to equal both 0 and 10 at the same time. Hence, the equations contradict each other, and no pair $(x, y)$ can satisfy both equations simultaneously.

As a result, the system of equations is inconsistent. Therefore, the correct solution is that there is no solution to the system, which corresponds to choice No solution.

Therefore, the final solution to the problem is No solution.

Final Answer

No solution

Key Points to Remember

Essential concepts to master this topic

Rule: Inconsistent systems have same left sides but different right sides
Technique: Compare coefficients: x+y=0 vs x+y=10 shows contradiction
Check: No values can make x+y equal both 0 and 10 simultaneously ✓

Common Mistakes

Avoid these frequent errors

Trying to solve by substitution or elimination
Don't attempt algebraic methods when equations contradict each other = wasted time and confusion! The different constants (0 vs 10) with identical left sides immediately show impossibility. Always check for contradictions before solving.

Practice Quiz

Test your knowledge with interactive questions

$\begin{cases} x+y=8 \\ x-y=6 \end{cases}$

FAQ

Everything you need to know about this question

How can I quickly spot an inconsistent system?

Look for equations with identical left sides but different right sides. Like $x+y=0$ and $x+y=10$ - same expression can't equal two different numbers!

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

No solution: Equations contradict (like $x+y=0$ and $x+y=10$ ). Infinite solutions: Equations are identical (like $x+y=0$ and $2x+2y=0$ ).

Should I graph this to verify there's no solution?

Great idea! Both equations represent parallel lines with the same slope but different y-intercepts. Parallel lines never intersect, confirming no solution exists.

What if I accidentally try to solve it anyway?

You'll get a false statement like $0 = 10$ . This contradiction confirms the system has no solution - it's actually a helpful verification method!

Are inconsistent systems common in real life?

Yes! Think of impossible situations: 'I need 5 apples and 5 apples costs $10' versus '5 apples costs $20'. These contradictory conditions have no solution.

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