Solve X and Y in System: 1/2x + 7/2y = 10, -3x + 7y = 12

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

Multiplying by 2 clears the fractions and makes the coefficients easier to work with. We get $ x + 7y = 20 $, which has the same y-coefficient as the second equation.

Q: Can I use substitution instead of elimination?

Yes! You could solve for $ x $ from the first equation: $ x = 20 - 7y $, then substitute into the second equation. Both methods work, but elimination is often faster for this type of system.

Q: What does y = 18/7 mean as a decimal?

$ \frac{18}{7} \approx 2.57 $ when rounded to two decimal places. Many systems have exact fractional answers that we approximate as decimals for practical use.

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

Look for coefficients that are already equal or easily made equal. Here, both equations have $ 7y $ terms after clearing fractions, making y the natural choice to eliminate.

Q: What if I get a negative answer when eliminating?

Negative results are normal! When we got $ -4x = -8 $, dividing both sides by -4 gives us the positive answer $ x = 2 $. Always divide carefully with negative coefficients.

System of Linear Equations with Mixed Coefficients

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

$\begin{cases} \frac{1}{2}x+\frac{7}{2}y=10 \\ -3x+7y=12 \end{cases}$

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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 Multiply by 2 to eliminate fractions

00:15 Now this is the system of equations

00:21 Subtract between the equations

00:29 Negative times negative always equals positive

00:32 Isolate X

00:40 This is the value of X, now substitute to find Y

00:49 Isolate Y

01:03 This is the value of Y

01:11 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.

$\begin{cases} \frac{1}{2}x+\frac{7}{2}y=10 \\ -3x+7y=12 \end{cases}$

Step-by-step solution

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

Step 1: Multiply the first equation to make the coefficients of $y$ equal to easily eliminate $y$ .
Step 2: Subtract one equation from the other to eliminate $y$ and solve for $x$ .
Step 3: Substitute the value of $x$ back into one of the original equations to solve for $y$ .

Now, let's work through each step:
Step 1: Multiply the first equation $\frac{1}{2}x + \frac{7}{2}y = 10$ by 2 to eliminate fractions:
$x + 7y = 20$

Step 2: Use the second equation as is: $-3x + 7y = 12$ . Subtract the equation $x + 7y = 20$ from $-3x + 7y = 12$ to eliminate $y$ :
$(-3x + 7y) - (x + 7y) = 12 - 20$
$-4x = -8$
Solve for $x$ :
$x = 2$

Step 3: Substitute $x = 2$ back into the equation $x + 7y = 20$ :
$2 + 7y = 20$
Subtract 2 from both sides:
$7y = 18$
Divide both sides by 7:
$y = \frac{18}{7} \approx 2.57$

Therefore, the solution that satisfies both equations is $(x, y) = (2, 2.57)$ .

The correct choice is $\boxed{x=2, y=2.57}$ .

Final Answer

$x=2,y=2.57$

Key Points to Remember

Essential concepts to master this topic

Elimination Method: Make coefficients equal by multiplying entire equations appropriately
Technique: Multiply $\frac{1}{2}x + \frac{7}{2}y = 10$ by 2 to get $x + 7y = 20$
Check: Substitute x = 2, y = 18/7 into both original equations ✓

Common Mistakes

Avoid these frequent errors

Only multiplying the fraction terms instead of the entire equation
Don't multiply just $\frac{1}{2}x$ and $\frac{7}{2}y$ by 2 while leaving 10 unchanged = unbalanced equation! This breaks the equality principle. Always multiply every term on both sides by the same number to maintain balance.

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 do we multiply the first equation by 2?

Multiplying by 2 clears the fractions and makes the coefficients easier to work with. We get $x + 7y = 20$ , which has the same y-coefficient as the second equation.

Can I use substitution instead of elimination?

Yes! You could solve for $x$ from the first equation: $x = 20 - 7y$ , then substitute into the second equation. Both methods work, but elimination is often faster for this type of system.

What does y = 18/7 mean as a decimal?

$\frac{18}{7} \approx 2.57$ when rounded to two decimal places. Many systems have exact fractional answers that we approximate as decimals for practical use.

How do I know which variable to eliminate first?

Look for coefficients that are already equal or easily made equal. Here, both equations have $7y$ terms after clearing fractions, making y the natural choice to eliminate.

What if I get a negative answer when eliminating?

Negative results are normal! When we got $-4x = -8$ , dividing both sides by -4 gives us the positive answer $x = 2$ . Always divide carefully with negative coefficients.

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