Solve the Linear Equation Pair: 6x + y = 12 and 3y + 2x = 20

Q: Why multiply the first equation by 3 instead of other numbers?

Multiplying by 3 creates matching coefficients for y: $ 3y $ in both equations. This lets us eliminate y completely in one step!

Q: Can I eliminate x instead of y?

Absolutely! You could multiply the second equation by 3 and the first by 1 to get matching x coefficients. The final answer will be the same: $ x=1, y=6 $.

Q: What does it mean to 'eliminate' a variable?

Eliminating means making a variable disappear by adding or subtracting equations. When coefficients are opposites (like +3y and -3y), they cancel out completely.

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

Substitute your values into both original equations:First: $ 6(1) + 6 = 12 $ ✓Second: $ 3(6) + 2(1) = 20 $ ✓

Q: What if I get different answers when I substitute back?

If substitution doesn't work, you made an arithmetic error. Go back and carefully check your multiplication and subtraction steps. The solution must satisfy both equations.

Linear Equations with Elimination Method

$6x+y=12$

$3y+2x=20$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:13 Let's multiply one of the equations by 3, so we can subtract between them

00:27 Now let's subtract between the equations

00:32 Let's simplify what we can

00:41 Collect like terms

00:47 Isolate X

00:52 This is the value of X

00:57 Now let's substitute X to find the value of Y

01:07 Isolate Y

01:14 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

$6x+y=12$

$3y+2x=20$

Step-by-step solution

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

Step 1: Align the equations for elimination. These equations are already simple, but you could multiply the first equation by 3 to align coefficients for $y$ .
Step 2: Multiply the first equation by 3 to facilitate elimination of $y$ :
$3(6x + y) = 3 \times 12$
Thus, we get:
$18x + 3y = 36$ .
Step 3: Rewrite and subtract the second equation from this new equation:
$18x + 3y - (3y + 2x) = 36 - 20$ ,
This simplifies to:
$16x = 16$ .
Step 4: Solve for $x$ :
$x = \frac{16}{16} = 1$ .
Step 5: Substitute $x = 1$ back into the first original equation to find $y$ :
$6(1) + y = 12$ ,
$6 + y = 12$ ,
$y = 12 - 6$ ,
$y = 6$ .

Therefore, the solution to the system of equations is $x = 1, y = 6$ .

Final Answer

$x=1,y=6$

Key Points to Remember

Essential concepts to master this topic

Alignment: Match coefficients by multiplying equations strategically
Elimination: Multiply first equation by 3: $18x + 3y = 36$
Verification: Substitute $x=1, y=6$ into both original equations ✓

Common Mistakes

Avoid these frequent errors

Incorrectly distributing when subtracting equations
Don't subtract $18x + 3y - 3y + 2x$ = wrong signs! This gives $20x = 16$ instead of $16x = 16$ . Always distribute the negative sign to every term: $18x + 3y - (3y + 2x)$ .

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 first equation by 3 instead of other numbers?

Multiplying by 3 creates matching coefficients for y: $3y$ in both equations. This lets us eliminate y completely in one step!

Can I eliminate x instead of y?

Absolutely! You could multiply the second equation by 3 and the first by 1 to get matching x coefficients. The final answer will be the same: $x=1, y=6$ .

What does it mean to 'eliminate' a variable?

Eliminating means making a variable disappear by adding or subtracting equations. When coefficients are opposites (like +3y and -3y), they cancel out completely.

How do I check if my solution is correct?

Substitute your values into both original equations:

First: $6(1) + 6 = 12$ ✓
Second: $3(6) + 2(1) = 20$ ✓

What if I get different answers when I substitute back?

If substitution doesn't work, you made an arithmetic error. Go back and carefully check your multiplication and subtraction steps. The solution must satisfy both equations.

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