Solve the System: 2x + y = 10 and x - y = 2 Step-by-Step

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

Choose the equation that makes solving for a variable easiest! In this problem, $ x - y = 2 $ is better because solving for $ x $ gives us $ x = y + 2 $ with no fractions.

Q: What if I get different answers when I check both original equations?

This means you made an error! Your solution must satisfy both equations simultaneously. Go back and check your algebra step-by-step, especially when expanding and combining like terms.

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

Absolutely! You could solve $ x - y = 2 $ for $ y $ to get $ y = x - 2 $, then substitute into the first equation. You'll get the same final answer!

Q: Why can't I just guess and check the multiple choice answers?

While that might work for this problem, understanding the method is crucial! Systems of equations appear in many real-world situations, and you need to know how to solve them systematically.

Q: What does the solution (4, 2) actually mean?

The solution $ (4, 2) $ means $ x = 4 $ and $ y = 2 $. This is the only point where both lines intersect when you graph these equations!

Systems of Linear Equations with Substitution Method

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

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

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

Step-by-step solution

To solve the system of equations:

$2x + y = 10$

$x - y = 2$

First, solve the second equation for $x$ :

$x = y + 2$

Substitute $x = y + 2$ into the first equation:

$2(y + 2) + y = 10$

Expand and simplify:

$2y + 4 + y = 10$

$3y + 4 = 10$

Subtract 4 from both sides:

$3y = 6$

Divide by 3:

$y = 2$

Substitute $y = 2$ back into $x = y + 2$ :

$x = 2 + 2 = 4$

Thus, the solution is:

$x = 4, y = 2$

Final Answer

$x = 4, y = 2$

Key Points to Remember

Essential concepts to master this topic

Substitution Rule: Solve one equation for a variable first
Technique: From $x - y = 2$ get $x = y + 2$
Check: Substitute $x = 4, y = 2$ back: $2(4) + 2 = 10$ ✓

Common Mistakes

Avoid these frequent errors

Adding equations without proper alignment of terms
Don't just add $2x + y + x - y = 10 + 2$ randomly = messy calculations! This leads to sign errors and wrong solutions. Always solve one equation for a single variable first, then substitute that expression into the other equation.

Practice Quiz

Test your knowledge with interactive questions

$\begin{cases} x+y=8 \\ x-y=6 \end{cases}$

FAQ

Everything you need to know about this question

Which equation should I solve for a variable first?

Choose the equation that makes solving for a variable easiest! In this problem, $x - y = 2$ is better because solving for $x$ gives us $x = y + 2$ with no fractions.

What if I get different answers when I check both original equations?

This means you made an error! Your solution must satisfy both equations simultaneously. Go back and check your algebra step-by-step, especially when expanding and combining like terms.

Can I solve for y first instead of x?

Absolutely! You could solve $x - y = 2$ for $y$ to get $y = x - 2$ , then substitute into the first equation. You'll get the same final answer!

Why can't I just guess and check the multiple choice answers?

While that might work for this problem, understanding the method is crucial! Systems of equations appear in many real-world situations, and you need to know how to solve them systematically.

What does the solution (4, 2) actually mean?

The solution $(4, 2)$ means $x = 4$ and $y = 2$ . This is the only point where both lines intersect when you graph these equations!

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