Detailed Equation Analysis: Decipher the System -2x+3y=4, x-4y=8

Q: Why multiply the second equation by 2 instead of the first?

We want to eliminate $ x $, so we need coefficients of $ +2x $ and $ -2x $. Since equation 1 has $ -2x $, multiplying equation 2 by 2 gives us $ +2x $ - perfect opposites!

Q: Could I eliminate y instead of x?

Absolutely! You could multiply equation 1 by 4 and equation 2 by 3 to get $ +12y $ and $ -12y $. Both methods work - choose whichever seems easier!

Q: How do I verify my solution works?

Substitute into both original equations:$ -2(-8) + 3(-4) = 16 - 12 = 4 $ ✓$ (-8) - 4(-4) = -8 + 16 = 8 $ ✓

System of Linear Equations with Elimination Method

Solve the above set of equations and choose the correct answer.

$\begin{cases} -2x+3y=4 \\ x-4y=8 \end{cases}$

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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 Isolate X

00:10 This is the expression of X using Y

00:17 Substitute this expression to find Y

00:30 Open parentheses properly, multiply by each factor

00:41 Isolate Y

00:48 Collect terms

01:01 Isolate Y

01:08 This is the value of Y

01:13 This is the value of Y

01:36 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

Solve the above set of equations and choose the correct answer.

$\begin{cases} -2x+3y=4 \\ x-4y=8 \end{cases}$

Step-by-step solution

To solve this problem, we'll follow these specific steps:

First, look at our system of equations:

Equation 1: $-2x + 3y = 4$
Equation 2: $x - 4y = 8$

We choose to use the elimination method to remove one variable from the equations. We'll aim to eliminate $x$ .
To achieve this, multiply the second equation by 2 so that we can align the coefficients of $x$ in both equations:

New Equation 2: $2x - 8y = 16$

Now, add the transformed second equation to Equation 1 to cancel out $x$ :

(-2x + 3y) + (2x - 8y) = 4 + 16

This simplifies to:

-5y = 20

Solve for $y$ :

y = -4

With $y$ known, substitute back into the second original equation to determine $x$ :

x - 4(-4) = 8

Simplify and solve for $x$ :

x + 16 = 8 \quad \Rightarrow \quad x = 8 - 16 \quad \Rightarrow \quad x = -8

We have now found the solution for the system of equations. The values are $x = -8$ and $y = -4$ .

Thus, the correct answer choice is $x = -8, y = -4$ .

Final Answer

$x=-8,y=-4$

Key Points to Remember

Essential concepts to master this topic

Rule: Align coefficients to eliminate one variable systematically
Technique: Multiply equation 2 by 2: $2(x - 4y) = 2(8)$
Check: Substitute $x = -8, y = -4$ into both original equations ✓

Common Mistakes

Avoid these frequent errors

Adding equations without aligning coefficients first
Don't add $-2x + 3y = 4$ and $x - 4y = 8$ directly = no variable cancels out! This leaves you with messy coefficients that don't eliminate. Always multiply one or both equations to make coefficients opposites before adding.

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

Why multiply the second equation by 2 instead of the first?

We want to eliminate $x$ , so we need coefficients of $+2x$ and $-2x$ . Since equation 1 has $-2x$ , multiplying equation 2 by 2 gives us $+2x$ - perfect opposites!

Could I eliminate y instead of x?

Absolutely! You could multiply equation 1 by 4 and equation 2 by 3 to get $+12y$ and $-12y$ . Both methods work - choose whichever seems easier!

What if I get different signs in my answer?

Sign errors are very common! Always be careful when multiplying negative numbers. Double-check: $x - 4(-4) = x + 16$ , not $x - 16$ .

How do I verify my solution works?

Substitute into both original equations:

$-2(-8) + 3(-4) = 16 - 12 = 4$ ✓
$(-8) - 4(-4) = -8 + 16 = 8$ ✓

Why did we get negative values for both variables?

The solution depends entirely on the given equations! Negative solutions are perfectly valid. The key is following the algebra correctly, regardless of whether answers are positive or negative.

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