Solve this System: 4x - 8y = 16 and -x - 2y = 24

Q: Why multiply the second equation by 4 instead of something else?

We want to eliminate one variable by making coefficients opposites. Since the first equation has $ 4x $ and the second has $ -x $, multiplying by 4 gives us $ -4x $, which cancels perfectly!

Q: What if I get confused about positive and negative signs?

Write each step carefully! When you have $ 4x + (-4x) $, they cancel to give 0. For $ -8y + (-8y) $, you're adding two negatives to get $ -16y $.

Q: Can I use substitution instead of elimination?

Yes! But elimination is often cleaner for this system because the coefficients work out nicely. With substitution, you'd need to solve $ x = -24 - 2y $ first, then substitute.

Q: How do I check if my solution is correct?

Substitute $ x = -10 $ and $ y = -7 $ into both original equations:First: $ 4(-10) - 8(-7) = -40 + 56 = 16 $ ✓Second: $ -(-10) - 2(-7) = 10 + 14 = 24 $ ✓

Q: What does it mean when both variables are negative?

Nothing wrong with that! The solution point $ (-10, -7) $ is in the third quadrant of the coordinate plane. Many systems have solutions with negative coordinates.

System of Equations with Elimination Method

$4x-8y=16$

$-x-2y=24$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solution

00:03 Let's multiply one of the equations by 4, so we can combine them

00:19 Now let's combine the equations

00:24 Let's reduce what we can

00:29 Let's group terms

00:37 Let's isolate Y

00:41 This is the value of Y

00:53 Now let's substitute Y to find the value of X

01:17 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$4x-8y=16$

$-x-2y=24$

Step-by-step solution

To solve this system using the elimination method, we'll follow these steps:

Step 1: Multiply the second equation by 4 to align the $x$ -coefficients:

The second equation is $-x - 2y = 24$ . Multiply it by 4:

$-4x - 8y = 96$

Step 2: Add this to the first equation:

The first equation is $4x - 8y = 16$ .

Adding the scaled second equation gives:

$(4x - 8y) + (-4x - 8y) = 16 + 96$

Solving gives:

$-16y = 112$

Step 3: Solve for $y$ :

Divide both sides by $-16$ :

$y = \frac{112}{-16} = -7$

Step 4: Substitute $y = -7$ into the original second equation:

Use $-x - 2y = 24$ :

$-x - 2(-7) = 24$

$-x + 14 = 24$

Subtract 14 from both sides:

$-x = 10$

Multiply by $-1$ :

$x = -10$

Thus, the solution to the system is $x = -10$ and $y = -7$ .

Therefore, the solution to the problem is $(x, y) = (-10, -7)$ .

Final Answer

$x=-10,y=-7$

Key Points to Remember

Essential concepts to master this topic

Rule: Align coefficients by multiplying entire equations strategically
Technique: Multiply second equation by 4: $-4x - 8y = 96$
Check: Substitute $x = -10, y = -7$ into both original equations ✓

Common Mistakes

Avoid these frequent errors

Multiplying only the coefficient instead of the entire equation
Don't multiply just the x-term by 4 and leave the constant unchanged = wrong equation! This destroys the balance and gives incorrect solutions. Always multiply every single term in the equation by the same number.

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

Why multiply the second equation by 4 instead of something else?

We want to eliminate one variable by making coefficients opposites. Since the first equation has $4x$ and the second has $-x$ , multiplying by 4 gives us $-4x$ , which cancels perfectly!

What if I get confused about positive and negative signs?

Write each step carefully! When you have $4x + (-4x)$ , they cancel to give 0. For $-8y + (-8y)$ , you're adding two negatives to get $-16y$ .

Can I use substitution instead of elimination?

Yes! But elimination is often cleaner for this system because the coefficients work out nicely. With substitution, you'd need to solve $x = -24 - 2y$ first, then substitute.

How do I check if my solution is correct?

Substitute $x = -10$ and $y = -7$ into both original equations:

First: $4(-10) - 8(-7) = -40 + 56 = 16$ ✓
Second: $-(-10) - 2(-7) = 10 + 14 = 24$ ✓

What does it mean when both variables are negative?

Nothing wrong with that! The solution point $(-10, -7)$ is in the third quadrant of the coordinate plane. Many systems have solutions with negative coordinates.

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