Solve the Equation: Use Substitution for 8x - 2y = 10 and 3x + 3y = 9

Q: Which equation should I solve for a variable first?

Choose the simpler equation! In this problem, $ 3x + 3y = 9 $ is easier because all coefficients are the same. Dividing by 3 gives you $ x + y = 3 $.

Q: How do I substitute a fraction correctly?

Be extra careful with parentheses! When substituting $ y = 3 - x $, write it as $ 8x - 2(3 - x) $. The parentheses ensure you distribute the -2 to both terms inside.

Q: What if my fractions look different from the answer choices?

Convert to the same form! $ \frac{7}{5} $ equals $ \frac{14}{10} $ because $ \frac{7 \times 2}{5 \times 2} = \frac{14}{10} $. Always check if your answer matches by cross-multiplying or finding equivalent fractions.

Q: Should I always check both equations?

Yes! Your solution must satisfy both original equations. If it only works in one equation, you made an error somewhere. Checking both equations confirms your answer is correct.

Substitution Method with Fractional Solutions

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

$\begin{cases} 8x-2y=10 \\ 3x+3y=9 \end{cases}$

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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

00:07 Simplify the equation as much as possible

00:22 Isolate X

00:27 This is the expression for X value, substitute in second equation to find Y

00:47 Open parentheses properly, multiply by each factor

00:56 Isolate Y

01:10 Combine terms

01:30 This is the Y value

01:36 Now substitute the Y value to find X

01:51 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

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

$\begin{cases} 8x-2y=10 \\ 3x+3y=9 \end{cases}$

Step-by-step solution

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

Solve $3x + 3y = 9$ for $y$ :

Divide the whole equation by 3 to simplify:
$x + y = 3$

Express $y$ in terms of $x$ :

$y = 3 - x$

Substitute $y = 3 - x$ into the first equation:

We substitute into $8x - 2y = 10$ :
$8x - 2(3 - x) = 10$

Simplify and solve for $x$ :

$8x - 6 + 2x = 10$
$10x - 6 = 10$
Add 6 to both sides:
$10x = 16$
Divide by 10:
$x = \frac{16}{10} = \frac{8}{5}$

Substitute back to find $y$ :

Use $y = 3 - x$ :
$y = 3 - \frac{8}{5} = \frac{15}{5} - \frac{8}{5} = \frac{7}{5}$

Therefore, the solution to the system is $x = \frac{8}{5}$ and $y = \frac{7}{5}$ .

Final Answer

$x=\frac{8}{5},y=\frac{14}{10}$

Key Points to Remember

Essential concepts to master this topic

Substitution Rule: Solve simpler equation first to express one variable
Technique: From $3x + 3y = 9$ , get $y = 3 - x$
Check: Substitute $x = \frac{8}{5}, y = \frac{7}{5}$ into both original equations ✓

Common Mistakes

Avoid these frequent errors

Substituting into the same equation used for isolation
Don't substitute $y = 3 - x$ back into $3x + 3y = 9$ = you get $9 = 9$ (always true)! This tells you nothing about x. Always substitute into the OTHER equation to find the variable value.

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

Which equation should I solve for a variable first?

Choose the simpler equation! In this problem, $3x + 3y = 9$ is easier because all coefficients are the same. Dividing by 3 gives you $x + y = 3$ .

Why did we get fractions as answers?

Fractional answers are completely normal in systems of equations! When coefficients don't divide evenly, you'll often get fractions. Just make sure to simplify and check your work.

How do I substitute a fraction correctly?

Be extra careful with parentheses! When substituting $y = 3 - x$ , write it as $8x - 2(3 - x)$ . The parentheses ensure you distribute the -2 to both terms inside.

What if my fractions look different from the answer choices?

Convert to the same form! $\frac{7}{5}$ equals $\frac{14}{10}$ because $\frac{7 \times 2}{5 \times 2} = \frac{14}{10}$ . Always check if your answer matches by cross-multiplying or finding equivalent fractions.

Should I always check both equations?

Yes! Your solution must satisfy both original equations. If it only works in one equation, you made an error somewhere. Checking both equations confirms your answer is correct.

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