Solving Systems of Equations: Substitution of Fractions in 2x-y/2 + -3y+x/5 = 7

Q: Why multiply the first equation by 10 and the second by 24?

These are the LCD values for each equation! First equation has denominators 2 and 5, so LCD = 10. Second equation has denominators 8 and 6, so LCD = 24. This clears all fractions efficiently.

Q: Can I solve for y first instead of x?

Absolutely! You can solve either equation for either variable first. Choose the variable that gives you the simplest expression to substitute.

Q: How do I handle the negative signs when clearing fractions?

Be extra careful with signs! When you have $ \frac{-3y+x}{5} $, multiplying by 2 gives -6y + 2x. Write out each step to avoid sign errors.

Q: Why are the final answers decimals instead of fractions?

The exact answers are $ x = \frac{157}{100} $ and $ y = -\frac{465}{100} $. Decimals like 1.57 and -4.65 are just the decimal form of these fractions.

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

Substitute your values back into both original equations with fractions. If $ \frac{2(1.57)-(-4.65)}{2}+\frac{-3(-4.65)+1.57}{5} = 7 $, you're correct!

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve the system of equations

00:16 Multiply by 10 to get rid of fractions

00:36 Open parentheses properly, multiply by each term

00:56 Collect terms

01:06 Isolate X

01:20 This is the value of X, we'll substitute it in the second equation to find Y

01:47 Multiply by 24 to get rid of fractions

02:06 Open parentheses properly, multiply by each term

02:34 Collect terms

02:48 Substitute the X value we found to find Y

03:00 Multiply by 12 to get rid of the fraction

03:23 Open parentheses properly, multiply by each term

03:38 Isolate Y

04:10 This is the value of Y

04:21 Now let's substitute the Y value to find X

05:13 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Solve the following equations for x and y using the substitution method:

$\begin{cases} \frac{2x-y}{2}+\frac{-3y+x}{5}=7 \\ \frac{-y-x}{8}-\frac{5y+x}{6}=4 \end{cases}$

2

Step-by-step solution

To solve the system of equations using substitution:

Step 1: Simplify each equation.

First equation:

\frac{2x - y}{2} + \frac{-3y + x}{5} = 7

Multiply by 10 to eliminate denominators:

(2x - y) \times 5 + (-3y + x) \times 2 = 70

10x - 5y - 6y + 2x = 70

12x - 11y = 70\quad\text{(1)}

Second equation:

\frac{-y - x}{8} - \frac{5y + x}{6} = 4

Multiply by 24 to eliminate denominators:

( -y - x) \times 3 - (5y + x) \times 4 = 96

-3y - 3x - 20y - 4x = 96

-7x - 23y = 96\quad\text{(2)}

Step 2: Solve the first equation for $x$ :

12x = 70 + 11y

x = \frac{70 + 11y}{12}\quad\text{(3)}

Step 3: Substitute Equation (3) into Equation (2):

-7\left(\frac{70 + 11y}{12}\right) - 23y = 96

Clear while multiplying by 12:

-7(70 + 11y) - 276y = 1152

-490 - 77y - 276y = 1152

-353y = 1642

y = -4.65

Step 4: Substitute $y = -4.65$ back into Equation (3) to find $x$ :

x = \frac{70 + 11(-4.65)}{12}

x = \frac{70 - 51.15}{12}

x = \frac{18.85}{12}

x = 1.57

Therefore, the solution to the problem is $x = 1.57$ , $y = -4.65$ .

3

Final Answer

$x=1.57,y=-4.65$

Key Points to Remember

Essential concepts to master this topic

Fraction Clearing: Multiply equations by LCD to eliminate all denominators completely
Substitution Method: Solve first equation for x: $x = \frac{70 + 11y}{12}$
Verification Check: Substitute $x = 1.57, y = -4.65$ into both original equations ✓

Common Mistakes

Avoid these frequent errors

Not finding common denominators before clearing fractions
Don't multiply equation 1 by 10 and equation 2 by 24 without planning = inconsistent clearing! This creates unnecessarily large numbers and increases calculation errors. Always find the LCD for each equation separately, then multiply every term by that LCD.

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 10 and the second by 24?

+

These are the LCD values for each equation! First equation has denominators 2 and 5, so LCD = 10. Second equation has denominators 8 and 6, so LCD = 24. This clears all fractions efficiently.

Can I solve for y first instead of x?

+

Absolutely! You can solve either equation for either variable first. Choose the variable that gives you the simplest expression to substitute.

How do I handle the negative signs when clearing fractions?

+

Be extra careful with signs! When you have $\frac{-3y+x}{5}$ , multiplying by 2 gives -6y + 2x. Write out each step to avoid sign errors.

Why are the final answers decimals instead of fractions?

+

The exact answers are $x = \frac{157}{100}$ and $y = -\frac{465}{100}$ . Decimals like 1.57 and -4.65 are just the decimal form of these fractions.

What if I get different simplified equations?

+

Double-check your arithmetic! The equations should simplify to $12x - 11y = 70$ and $-7x - 23y = 96$ . Any error here will give wrong final answers.

How can I check if my solution is correct?

+

Substitute your values back into both original equations with fractions. If $\frac{2(1.57)-(-4.65)}{2}+\frac{-3(-4.65)+1.57}{5} = 7$ , you're correct!

🌟 Unlock Your Math Potential

Get unlimited access to all 18 System of linear equations questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime

Solving Systems of Equations: Substitution of Fractions in 2x-y/2 + -3y+x/5 = 7

Linear Systems with Fractional Simplification

❤️ Continue Your Math Journey!