Solve This Equation System: 2x - 1/5y = 18 and 3x + y = 6

Q: Why multiply the first equation by 5 instead of working with the fraction?

Multiplying by 5 eliminates the fraction $-\frac{1}{5}y$ completely, making it easier to use elimination. Working with fractions throughout the process is much more error-prone and time-consuming.

Q: How do I know which variable to eliminate?

Look for the easiest elimination! After clearing fractions, we have -y and +y, which add to zero perfectly. This makes y the obvious choice to eliminate first.

Q: My answer has decimals - is that wrong?

Not at all! $x = \frac{96}{13}$ gives exactly 7.384615..., so x = 7.38 is the correct rounded answer. Always check if the problem asks for exact fractions or decimal approximations.

Q: How can I verify my solution is correct?

Substitute both values into both original equations:Equation 1: $2(7.38) - \frac{1}{5}(-16.14) = 14.76 + 3.228 = 17.988 ≈ 18$ ✓Equation 2: $3(7.38) + (-16.14) = 22.14 - 16.14 = 6$ ✓

System of Equations with Fractional Coefficients

Solve the following system of equations:

$\begin{cases} 2x-\frac{1}{5}y=18 \\ 3x+y=6 \end{cases}$

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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

00:04 Multiply by 5 to eliminate the fraction

00:15 Now this is the system of equations

00:21 Add the equations

00:34 Collect like terms

00:46 Isolate X

00:55 This is the value of X, now substitute to find Y

01:16 Isolate Y

01:27 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

Solve the following system of equations:

$\begin{cases} 2x-\frac{1}{5}y=18 \\ 3x+y=6 \end{cases}$

Step-by-step solution

To solve the given system of equations using elimination, we'll follow these steps:

Step 1: Simplify the first equation to remove the fraction.
Step 2: Make the coefficients of $y$ in both equations equal, to facilitate elimination.
Step 3: Eliminate $y$ by subtracting the equations.
Step 4: Solve for $x$ .
Step 5: Use the value of $x$ to find the value of $y$ .

Step 1: Multiply the first equation by 5 to clear the fraction:

$10x - y = 90$

Step 2: The second equation is already in a suitable form for elimination:

$3x + y = 6$

Step 3: Add the two equations:

$(10x - y) + (3x + y) = 90 + 6$

This simplifies to:

$13x = 96$

Step 4: Solve for $x$ :

$x = \frac{96}{13} = 7.38$

Step 5: Substitute $x = 7.38$ back into the second equation to find $y$ :

Edit Form|li $3(7.38) + y = 6$

$22.14 + y = 6$

$y = 6 - 22.14$

$y = -16.14$

Therefore, the solution to the system of equations is $x = 7.38$ , $y = -16.14$ .

Final Answer

$x=7.38,y=-16.14$

Key Points to Remember

Essential concepts to master this topic

Rule: Multiply first equation by 5 to eliminate the fraction
Technique: Use elimination: 10x - y = 90 and 3x + y = 6
Check: Substitute x = 7.38, y = -16.14 back into both original equations ✓

Common Mistakes

Avoid these frequent errors

Forgetting to multiply ALL terms by 5
Don't just multiply -1/5y by 5 = y and leave 2x - 18 unchanged! This creates an incorrect equation 2x - y = 18 instead of 10x - y = 90. Always multiply every single term in the equation by the same number.

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 first equation by 5 instead of working with the fraction?

Multiplying by 5 eliminates the fraction $-\frac{1}{5}y$ completely, making it easier to use elimination. Working with fractions throughout the process is much more error-prone and time-consuming.

How do I know which variable to eliminate?

Look for the easiest elimination! After clearing fractions, we have -y and +y, which add to zero perfectly. This makes y the obvious choice to eliminate first.

Can I use substitution instead of elimination?

Yes, but elimination is faster here! With substitution, you'd solve $y = 6 - 3x$ from equation 2, then substitute into the messy fractional equation 1. Elimination avoids fractions entirely.

My answer has decimals - is that wrong?

Not at all! $x = \frac{96}{13}$ gives exactly 7.384615..., so x = 7.38 is the correct rounded answer. Always check if the problem asks for exact fractions or decimal approximations.

How can I verify my solution is correct?

Substitute both values into both original equations:

Equation 1: $2(7.38) - \frac{1}{5}(-16.14) = 14.76 + 3.228 = 17.988 ≈ 18$ ✓
Equation 2: $3(7.38) + (-16.14) = 22.14 - 16.14 = 6$ ✓

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