Solve the Equations: 3x-y=5 and 5x+2y=12 for x and y

Q: Why do we multiply the first equation by 2?

We multiply by 2 to make the y-coefficients opposites (-2y and +2y). This way, when we add the equations together, the y-terms cancel out completely, leaving us with just x!

Q: What if I get different answers when I check?

If your solution doesn't work in both original equations, you made an error. Go back and check your arithmetic, especially when multiplying equations and combining like terms.

Q: How do I know which variable to eliminate first?

Look for the easier path! In this problem, eliminating y only required multiplying by 2, while eliminating x would need bigger numbers. Choose the simpler option.

Q: What does the solution (2,1) actually mean?

The solution $ (2,1) $ means x = 2 and y = 1. This is the only point where both lines intersect - it's the one pair of values that satisfies both equations simultaneously.

System of Linear Equations with Elimination Method

$3x-y=5$

$5x+2y=12$

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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 combine them

00:16 Now let's combine the equations

00:21 Let's simplify what we can

00:35 Let's group like terms

00:39 Let's isolate X

00:44 This is the value of X

00:49 Now let's substitute X to find the value of Y

01:03 Let's isolate Y

01:12 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

$3x-y=5$

$5x+2y=12$

Step-by-step solution

To solve the system of equations using the elimination method, follow these steps:

Step 1: Write down the given equations:
$3x - y = 5$ (Equation 1)
$5x + 2y = 12$ (Equation 2)
Step 2: Align coefficients for elimination. Here, we aim to eliminate $y$ by aligning coefficients of $y$ in both equations. Multiply Equation 1 by $2$ :
$2(3x - y) = 2(5)$ , which simplifies to $6x - 2y = 10$ .
Step 3: Add the modified Equation 1 to Equation 2:
$(6x - 2y) + (5x + 2y) = 10 + 12$ .
The $2y$ terms cancel out, leaving $11x = 22$ .
Step 4: Solve for $x$ :
$11x = 22$ simplifies to $x = 2$ .
Step 5: Substitute $x = 2$ back into Equation 1 to solve for $y$ :
$3(2) - y = 5$ , which simplifies to $6 - y = 5$ . Thus, $y = 1$ .

Therefore, the solution to the system of equations is $x = 2$ and $y = 1$ .

This corresponds to choice 2 in the provided answer choices.

Final Answer

$x=2,y=1$

Key Points to Remember

Essential concepts to master this topic

System Setup: Write both equations clearly before starting elimination
Elimination Technique: Multiply first equation by 2: $6x - 2y = 10$
Verification: Substitute $x = 2, y = 1$ into both original equations ✓

Common Mistakes

Avoid these frequent errors

Forgetting to multiply all terms when aligning coefficients
Don't multiply just the y-term by 2 and leave 3x unchanged = wrong equation! This creates an invalid equation that doesn't represent the original relationship. 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 do we multiply the first equation by 2?

We multiply by 2 to make the y-coefficients opposites (-2y and +2y). This way, when we add the equations together, the y-terms cancel out completely, leaving us with just x!

Can I eliminate x instead of y?

Absolutely! You could multiply the first equation by 5 and the second by -3 to get opposite x-coefficients. Both methods work - choose whichever looks easier to you.

What if I get different answers when I check?

If your solution doesn't work in both original equations, you made an error. Go back and check your arithmetic, especially when multiplying equations and combining like terms.

How do I know which variable to eliminate first?

Look for the easier path! In this problem, eliminating y only required multiplying by 2, while eliminating x would need bigger numbers. Choose the simpler option.

What does the solution (2,1) actually mean?

The solution $(2,1)$ means x = 2 and y = 1. This is the only point where both lines intersect - it's the one pair of values that satisfies both equations simultaneously.

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