Substitution Success: Solve for x and y in x + 4y = 5 - 3y, 2x + 3y = 6

Q: Why do I need to move the -3y to the left side first?

Moving all y-terms to one side makes the equation cleaner! Instead of working with x + 4y = 5 - 3y, you get the simpler form x + 7y = 5, which is much easier to solve for x.

Q: How do I handle fractions when substituting back?

Work step by step! When substituting $ y = \frac{4}{11} $, calculate $ 7 \times \frac{4}{11} = \frac{28}{11} $ first, then subtract: $ x = 5 - \frac{28}{11} = \frac{55}{11} - \frac{28}{11} = \frac{27}{11} $

Q: Should I always get fractional answers in systems?

Not always! Some systems have integer solutions, others have fractions. The answer depends on the specific equations. Fractional answers are completely normal and correct when simplified properly.

Q: How can I check if my fractional answer is right?

Substitute both values back into both original equations. For example: Check $ \frac{27}{11} + 4(\frac{4}{11}) $ equals $ 5 - 3(\frac{4}{11}) $, and verify the second equation too!

Q: What if I get confused with all the fraction arithmetic?

Take it one step at a time! Convert whole numbers to fractions with the same denominator, then add or subtract numerators. Write out each step clearly to avoid mistakes.

Systems of Equations with Fractional Solutions

Find the value of x and and band the substitution method.

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

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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 Isolate X

00:15 Collect terms

00:18 This is the X value expression, substitute in the second equation to find Y

00:41 Open parentheses properly, multiply by each factor

00:49 Isolate Y

01:13 This is the Y value

01:17 Now substitute Y value to find X

01:42 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

Find the value of x and and band the substitution method.

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

Step-by-step solution

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

Step A: Simplify and solve the first equation for $x$ .
Given: $x + 4y = 5 - 3y$
Combine like terms: $x + 4y + 3y = 5$ $x + 7y = 5$
Now solve for $x$ : $x = 5 - 7y$
Step B: Substitute this expression for $x$ in the second equation: $2x + 3y = 6$
Substitute $x = 5 - 7y$ : $2(5 - 7y) + 3y = 6$
Expand and simplify: $10 - 14y + 3y = 6$ $10 - 11y = 6$ $-11y = 6 - 10$ $-11y = -4$
Solving for $y$ : $y = \frac{-4}{-11} = \frac{4}{11}$
Step C: Substitute $y = \frac{4}{11}$ back into the equation for $x$ : $x = 5 - 7(\frac{4}{11})$ $x = 5 - \frac{28}{11}$
Convert 5 to an equivalent fraction: $x = \frac{55}{11} - \frac{28}{11}$ $x = \frac{27}{11}$

The solution to the system of equations is $x = \frac{27}{11}$ and $y = \frac{4}{11}$ .

Final Answer

$x=\frac{27}{11},y=\frac{4}{11}$

Key Points to Remember

Essential concepts to master this topic

Simplification: Combine like terms before substituting variables
Substitution: Express x = 5 - 7y then substitute into second equation
Verification: Check both solutions in original equations: $\frac{27}{11} + 4(\frac{4}{11}) = 5 - 3(\frac{4}{11})$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to combine like terms in the first equation
Don't substitute x + 4y = 5 - 3y directly without simplifying = creates complex fractions and errors! The 4y and -3y terms must be combined first. Always simplify x + 4y + 3y = 5 to get x + 7y = 5 before substituting.

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 I need to move the -3y to the left side first?

Moving all y-terms to one side makes the equation cleaner! Instead of working with x + 4y = 5 - 3y, you get the simpler form x + 7y = 5, which is much easier to solve for x.

How do I handle fractions when substituting back?

Work step by step! When substituting $y = \frac{4}{11}$ , calculate $7 \times \frac{4}{11} = \frac{28}{11}$ first, then subtract: $x = 5 - \frac{28}{11} = \frac{55}{11} - \frac{28}{11} = \frac{27}{11}$

Should I always get fractional answers in systems?

Not always! Some systems have integer solutions, others have fractions. The answer depends on the specific equations. Fractional answers are completely normal and correct when simplified properly.

How can I check if my fractional answer is right?

Substitute both values back into both original equations. For example: Check $\frac{27}{11} + 4(\frac{4}{11})$ equals $5 - 3(\frac{4}{11})$ , and verify the second equation too!

What if I get confused with all the fraction arithmetic?

Take it one step at a time! Convert whole numbers to fractions with the same denominator, then add or subtract numerators. Write out each step clearly to avoid mistakes.

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