Solve the Equation System: 2x - 2y = 10 and 4x - 4y = 32

Q: How can I quickly tell if a system has no solution?

Simplify both equations to their simplest form. If you get the same variable expression on the left but different numbers on the right (like $ x - y = 5 $ and $ x - y = 8 $), there's no solution!

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

No solution: Parallel lines that never meet (different constants). Infinite solutions: The exact same line written differently (same constants after simplifying).

Q: Why does x - y = 5 and x - y = 8 mean no solution?

Think about it: you can't have $ x - y $ equal to both 5 and 8 at the same time! That's impossible, so there's no pair of (x,y) values that satisfies both equations.

Q: Should I still try substitution if I think there's no solution?

If you've simplified and see the contradiction, stop there! Attempting substitution will just lead to false statements like 5 = 8, confirming what you already know.

Q: How do I write 'no solution' in math notation?

You can write it as 'No solution' or use the symbol $ \emptyset $ (empty set). Both are correct ways to indicate the system is inconsistent.

Linear Systems with Inconsistent Equations

$2x-2y=10$

$4x-4y=32$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:03 Let's multiply one of the equations by 2, so we can subtract between them

00:19 Now let's subtract between the equations

00:33 Let's collect like terms

00:46 We got an illogical expression, therefore there is no solution

00:56 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

$2x-2y=10$

$4x-4y=32$

Step-by-step solution

We start by analyzing the given system of equations:

The first equation is $2x - 2y = 10$ .

The second equation is $4x - 4y = 32$ .

To determine the relationship between these two lines, let's simplify both equations.

1. Simplify the first equation:
Divide every term by 2:
$x - y = 5$ .

2. Simplify the second equation:
Divide every term by 4:
$x - y = 8$ .

Notice that after simplification, we have:

First equation: $x - y = 5$
Second equation: $x - y = 8$

Upon comparison, both equations simplify to lines with the same slope but different intercepts. Therefore, they represent two parallel lines that do not intersect.

Consequently, the system of equations has no solution since parallel lines never meet.

Therefore, the correct answer is: No solution.

Final Answer

No solution

Key Points to Remember

Essential concepts to master this topic

Parallel Lines: Same slope, different intercepts means no intersection point
Technique: Simplify both equations: $x - y = 5$ vs $x - y = 8$
Check: If simplified equations have same left side but different right sides, no solution ✓

Common Mistakes

Avoid these frequent errors

Assuming all linear systems have solutions
Don't automatically try to solve by substitution or elimination without checking first = wasted time and confusion! Parallel lines never intersect, so some systems have no solution. Always simplify both equations first to compare their simplified forms.

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 tell if a system has no solution?

Simplify both equations to their simplest form. If you get the same variable expression on the left but different numbers on the right (like $x - y = 5$ and $x - y = 8$ ), there's no solution!

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

No solution: Parallel lines that never meet (different constants). Infinite solutions: The exact same line written differently (same constants after simplifying).

Why does x - y = 5 and x - y = 8 mean no solution?

Think about it: you can't have $x - y$ equal to both 5 and 8 at the same time! That's impossible, so there's no pair of (x,y) values that satisfies both equations.

Should I still try substitution if I think there's no solution?

If you've simplified and see the contradiction, stop there! Attempting substitution will just lead to false statements like 5 = 8, confirming what you already know.

How do I write 'no solution' in math notation?

You can write it as 'No solution' or use the symbol $\emptyset$ (empty set). Both are correct ways to indicate the system is inconsistent.

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