Leverage the System of Equations: 'x - y = 8' and '2x - 2y = 16' to Find x and y

Q: How can there be infinite solutions? Don't equations always have one answer?

Not always! When two equations represent the same line, every point on that line is a solution. For example, (10,2), (9,1), and (0,-8) all satisfy $ x - y = 8 $.

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

Infinite solutions: equations become identical (like $ x - y = 8 $ and $ x - y = 8 $). No solution: you get a false statement like $ 0 = 5 $.

Q: Why does dividing the second equation by 2 help?

Dividing by 2 simplifies the equation to its most basic form. This reveals whether it's the same as the first equation. Always simplify to compare equations clearly!

Q: Can I write the infinite solutions in a different way?

Yes! Since $ x - y = 8 $, you can write $ y = x - 8 $. Any value you choose for x gives you the corresponding y value.

Q: How do I know which method to use for systems of equations?

Start by simplifying both equations first. If they become identical, you have infinite solutions. If you get a contradiction, there's no solution. Otherwise, use substitution or elimination!

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

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

00:14 Now let's subtract between the equations

00:18 Let's simplify what we can

00:26 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 problem, we'll follow these steps:

Step 1: Write the given equations.
Step 2: Simplify and compare the equations to each other.
Step 3: Determine the nature of the solution.

Now, let's work through each step:

Step 1: The given system of equations is:

$x - y = 8$
$2x - 2y = 16$

Step 2: Simplify and compare the two equations:

The second equation $2x - 2y = 16$ can be divided entirely by 2 to give:

$x - y = 8$

Step 3: Observe that both equations are identical, meaning they represent the same line.

Therefore, this system of equations has infinite solutions, as every point on the line satisfies the equation.

The correct answer to the original problem is: Infinite solutions

Final Answer

Infinite solutions

Key Points to Remember

Essential concepts to master this topic

Recognition: Identical equations after simplification indicate infinite solutions
Technique: Divide second equation by 2: $2x - 2y = 16$ becomes $x - y = 8$
Check: Both equations represent same line, so every point satisfies both ✓

Common Mistakes

Avoid these frequent errors

Trying to solve for specific x and y values
Don't attempt to find exact numbers like x=5, y=-3 when equations are identical = wasted effort and confusion! When simplified equations match exactly, they represent the same line. Always recognize that identical equations mean infinite solutions along that line.

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 there be infinite solutions? Don't equations always have one answer?

Not always! When two equations represent the same line, every point on that line is a solution. For example, (10,2), (9,1), and (0,-8) all satisfy $x - y = 8$ .

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

Infinite solutions: equations become identical (like $x - y = 8$ and $x - y = 8$ ). No solution: you get a false statement like $0 = 5$ .

Why does dividing the second equation by 2 help?

Dividing by 2 simplifies the equation to its most basic form. This reveals whether it's the same as the first equation. Always simplify to compare equations clearly!

Can I write the infinite solutions in a different way?

Yes! Since $x - y = 8$ , you can write $y = x - 8$ . Any value you choose for x gives you the corresponding y value.

How do I know which method to use for systems of equations?

Start by simplifying both equations first. If they become identical, you have infinite solutions. If you get a contradiction, there's no solution. Otherwise, use substitution or elimination!

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