Solve for X and Y: Exploring the System -8x + 5y = 2, -16x + 10y = 5

Q: How can I tell if a system has no solution just by looking at it?

Look for proportional coefficients with different constant ratios! If you can multiply one equation to get the same left side as another equation, but the right sides don't match, there's no solution.

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

No solution: The lines are parallel (same slope, different y-intercepts). Infinite solutions: The equations represent the same line (everything is proportional).

Q: Why do we get 0 = 1 when there's no solution?

When you eliminate variables and get a false statement like $0 = 1$, it means the original equations contradict each other. This is mathematical proof that no point can satisfy both equations!

Q: Can I still use substitution method for this type of problem?

Yes! Substitution will also lead to a contradiction like $2 = 5$. Any valid method will reveal the inconsistency, but elimination often makes it clearer.

Q: How is this different from solving regular linear equations?

Regular linear equations always have one solution. Systems can have one solution, no solution, or infinite solutions. Always check for these three possibilities!

Q: What should I write as my final answer?

Simply write "No solution" or use the symbol $\emptyset$ (empty set). Don't try to find x and y values - they don't exist for inconsistent systems!

System of Equations with No Solution

Choose the correct answer for the following exercise:

$\begin{cases} -8x+5y=2 \\ -16x+10y=5 \end{cases}$

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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

00:03 Multiply by 2 so that we can isolate Y by subtraction

00:17 Now this is the system of equations

00:27 Subtract between the equations

00:45 Collect like terms

00:55 We got an illogical expression, therefore there is no solution

01:00 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

Choose the correct answer for the following exercise:

$\begin{cases} -8x+5y=2 \\ -16x+10y=5 \end{cases}$

Step-by-step solution

To solve this system of equations, we'll follow these steps:

Step 1: Analyze the proportions of the coefficients to check for potential parallelism or redundancy.
Step 2: Attempt elimination to directly identify any inconsistencies.

Now, let's perform the analysis:
Step 1: Notice the proportionality in both equations:
The first equation is $-8x + 5y = 2$ and the second is $-16x + 10y = 5$ . Multiply the first equation by 2:
$-16x + 10y = 4$ .
This equation is now equivalent in terms of $x$ and $y$ to the second equation, but with a different constant term.

Step 2: Subtract the modified first equation from the second equation:

(-16x + 10y) - (-16x + 10y) = 5 - 4

0 = 1

This result, $0 = 1$ , is a contradiction, indicating that the system is inconsistent.

Therefore, the system of equations has no solution.

Final Answer

No solution

Key Points to Remember

Essential concepts to master this topic

Inconsistency Check: When coefficients are proportional but constants aren't, no solution exists
Technique: Multiply first equation by 2: $-16x + 10y = 4$ vs $-16x + 10y = 5$
Verification: Subtracting identical left sides gives $0 = 1$ , confirming inconsistency ✓

Common Mistakes

Avoid these frequent errors

Assuming parallel lines always mean infinite solutions
Don't think that when coefficients are proportional you automatically get infinite solutions = wrong conclusion! Proportional coefficients with different constants create parallel lines that never intersect. Always check if the constant terms are also proportional to determine if there are infinite solutions or no solution.

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

How can I tell if a system has no solution just by looking at it?

Look for proportional coefficients with different constant ratios! If you can multiply one equation to get the same left side as another equation, but the right sides don't match, there's no solution.

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

No solution: The lines are parallel (same slope, different y-intercepts). Infinite solutions: The equations represent the same line (everything is proportional).

Why do we get 0 = 1 when there's no solution?

When you eliminate variables and get a false statement like $0 = 1$ , it means the original equations contradict each other. This is mathematical proof that no point can satisfy both equations!

Can I still use substitution method for this type of problem?

Yes! Substitution will also lead to a contradiction like $2 = 5$ . Any valid method will reveal the inconsistency, but elimination often makes it clearer.

How is this different from solving regular linear equations?

Regular linear equations always have one solution. Systems can have one solution, no solution, or infinite solutions. Always check for these three possibilities!

What should I write as my final answer?

Simply write "No solution" or use the symbol $\emptyset$ (empty set). Don't try to find x and y values - they don't exist for inconsistent systems!

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