Algebra Challenge: Solve the System -8x + 5y = 10 and -24x + 15y = 30

Q: How do I know if equations are dependent?

Check if the coefficients have the same ratio. In this problem: $ \frac{-24}{-8} = 3 $ and $ \frac{15}{5} = 3 $ and $ \frac{30}{10} = 3 $. All ratios are equal!

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

No solution: When you get something impossible like 0 = 5 after simplifying. Infinite solutions: When equations are the same line (dependent), so every point on the line works!

Q: Why can't I just solve this normally with substitution?

You can try, but you'll end up with 0 = 0, which means the equations are the same. This tells you there are infinite solutions, not that you made an error!

Q: How do I write the infinite solutions?

Express one variable in terms of the other. From $ -8x + 5y = 10 $, you get $ y = \frac{8x + 10}{5} $. Any x-value gives a corresponding y-value that works!

Q: Can I check my answer by picking any point?

Yes! Pick any x-value, find the corresponding y using $ y = \frac{8x + 10}{5} $, then verify both original equations are satisfied. Try $ x = 0 $: $ y = 2 $.

System of Equations with Dependent Equations

Choose the correct answer for the following exercise:

$\begin{cases} -8x+5y=10 \\ -24x+15y=30 \end{cases}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:11 Alright, let's solve this system of equations together.

00:17 First, multiply each term by 3. This helps us isolate Y by subtracting later.

00:28 Okay, here is our system of equations now.

00:38 Next, let's subtract one equation from the other.

00:53 Great! Now, let's collect like terms to simplify.

01:05 This tells us that X and Y work for any value. Awesome, right?

01:28 And there you have it! That's how we solve this 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:

$\begin{cases} -8x+5y=10 \\ -24x+15y=30 \end{cases}$

Step-by-step solution

To solve this problem using the substitution method, we'll carefully examine the structure of the given system of equations:

Given equations:
1) $-8x + 5y = 10$
2) $-24x + 15y = 30$

Notice that the second equation is exactly three times the first equation:

-24x + 15y = 3(-8x + 5y) = 3 \times 10 = 30

This implies the two equations are not independent; rather, they are multiples of each other.

This insight tells us that every solution of the first equation is also a solution of the second equation, which means:

The system has infinitely many solutions.

Given this conclusion, when examining the choices provided, the correct choice is "Infinite solutions."

Therefore, the solution to the system of equations is that it has infinite solutions.

Final Answer

Infinite solutions

Key Points to Remember

Essential concepts to master this topic

Dependent Equations: When one equation is a multiple of another
Recognition Method: Check if coefficients have same ratio: -24/-8 = 15/5 = 30/10
Verification: Multiply first equation by 3 to get second equation exactly ✓

Common Mistakes

Avoid these frequent errors

Attempting to solve dependent equations as if they were independent
Don't try to eliminate variables when equations are multiples of each other = wasted time and confusion! This leads to 0 = 0, which students think means no solution. Always check if one equation is a multiple of another first to identify infinite solutions.

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 do I know if equations are dependent?

Check if the coefficients have the same ratio. In this problem: $\frac{-24}{-8} = 3$ and $\frac{15}{5} = 3$ and $\frac{30}{10} = 3$ . All ratios are equal!

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

No solution: When you get something impossible like 0 = 5 after simplifying. Infinite solutions: When equations are the same line (dependent), so every point on the line works!

Why can't I just solve this normally with substitution?

You can try, but you'll end up with 0 = 0, which means the equations are the same. This tells you there are infinite solutions, not that you made an error!

How do I write the infinite solutions?

Express one variable in terms of the other. From $-8x + 5y = 10$ , you get $y = \frac{8x + 10}{5}$ . Any x-value gives a corresponding y-value that works!

Can I check my answer by picking any point?

Yes! Pick any x-value, find the corresponding y using $y = \frac{8x + 10}{5}$ , then verify both original equations are satisfied. Try $x = 0$ : $y = 2$ .

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