Solve the Linear Equation Pair: x-y=8 and 2x-2y=16

Q: How do I know when a system has infinite solutions?

A system has infinite solutions when the equations are dependent - meaning they represent the same line. After simplifying, if you get identical equations like $ x - y = 8 $, then every point on that line is a solution!

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

No solution: Equations are parallel lines that never intersect (like $ x - y = 8 $ and $ x - y = 3 $)Infinite solutions: Equations represent the same exact line, so they overlap completely.

Q: Can I still write specific solutions for infinite solutions?

Yes! You can express solutions as parametric equations. For $ x - y = 8 $, you could write: $ x = t, y = t - 8 $ where t is any real number.

Q: Why did dividing the second equation by 2 work?

Dividing an entire equation by a non-zero constant doesn't change the solution set! It's like saying $ 6 = 6 $ is the same as $ 3 = 3 $ when you divide both sides by 2.

Q: How do I graph a system with infinite solutions?

Both equations graph as the same line! Draw the line $ x - y = 8 $ and you'll see that every point on this line satisfies both original equations.

Linear Systems with Dependent Equations

$x-y=8$

$2x-2y=16$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:04 Let's multiply one of the equations by 2, so we can subtract between them

00:18 Now let's subtract between the equations

00:24 Let's reduce what we can

00:37 There are infinite solutions

00:41 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

$x-y=8$

$2x-2y=16$

Step-by-step solution

To solve this system of equations, let's first look at the given equations:
Equation 1: $x - y = 8$
Equation 2: $2x - 2y = 16$

Let's simplify the second equation. We can divide the entire equation by 2:

$\frac{2x - 2y}{2} = \frac{16}{2}$

This simplifies to:
$x - y = 8$

We can see now that both equations are identical:
1. $x - y = 8$
2. $x - y = 8$

Since both equations represent the same line, every point that is a solution to the first equation is also a solution to the second equation. This means that there are infinitely many solutions, as every point on the line $x - y = 8$ is a solution.

Therefore, the system of equations has infinite solutions.

Final Answer

Infinite solutions

Key Points to Remember

Essential concepts to master this topic

Dependent Equations: When equations simplify to identical forms, infinite solutions exist
Simplification: Divide $2x - 2y = 16$ by 2 to get $x - y = 8$
Verification: Both equations represent the same line on coordinate plane ✓

Common Mistakes

Avoid these frequent errors

Trying to solve dependent equations as if they're independent
Don't attempt substitution or elimination when equations are identical = wasting time and confusion! This leads to statements like 0 = 0 which students think means no solution. Always first check if equations simplify to the same form to identify infinite solutions.

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 do I know when a system has infinite solutions?

A system has infinite solutions when the equations are dependent - meaning they represent the same line. After simplifying, if you get identical equations like $x - y = 8$ , then every point on that line is a solution!

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

No solution: Equations are parallel lines that never intersect (like $x - y = 8$ and $x - y = 3$ )
Infinite solutions: Equations represent the same exact line, so they overlap completely.

Can I still write specific solutions for infinite solutions?

Yes! You can express solutions as parametric equations. For $x - y = 8$ , you could write: $x = t, y = t - 8$ where t is any real number.

Why did dividing the second equation by 2 work?

Dividing an entire equation by a non-zero constant doesn't change the solution set! It's like saying $6 = 6$ is the same as $3 = 3$ when you divide both sides by 2.

How do I graph a system with infinite solutions?

Both equations graph as the same line! Draw the line $x - y = 8$ and you'll see that every point on this line satisfies both original equations.

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