Solve the System of Equations: 5x+y=21 and x-y=3

Q: Why solve for x in the second equation instead of the first?

The second equation $ x - y = 3 $ is simpler to solve because the coefficient of x is 1. This avoids fractions and makes substitution easier!

Q: What if I solved for y instead of x first?

That works too! You'd get $ y = x - 3 $, then substitute into the first equation. You'll get the same answer: x = 4, y = 1.

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

Substitute both values into both original equations:$ 5(4) + 1 = 21 $ ✓$ 4 - 1 = 3 $ ✓If both check out, your solution is correct!

Q: Can I use elimination instead of substitution?

Yes! For this system, you could add the equations directly since the y terms are opposites. Both methods give the same answer, so use whichever feels easier to you.

Q: What does the solution (4, 1) mean graphically?

The point $ (4, 1) $ is where the two lines intersect on a coordinate plane. It's the only point that satisfies both equations simultaneously!

System of Equations with Substitution Method

$\begin{cases}5x+y=21 \\ x-y=3\end{cases}$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$\begin{cases}5x+y=21 \\ x-y=3\end{cases}$

Step-by-step solution

To solve the system of equations:

$5x+y=21$

$x - y = 3$

Let's solve the second equation for $x$ :

$x = y + 3$

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

$5(y + 3) + y = 21$

Simplify:

$5y + 15 + y = 21$

$6y = 6$

$y = 1$

Now, substitute $y = 1$ back into $x = y + 3$ :

$x = 1 + 3$

$x = 4$

Thus, the solution is $x = 4$ , $y = 1$ .

Final Answer

$x = 4$ , $y = 1$

Key Points to Remember

Essential concepts to master this topic

Rule: Solve one equation for a variable first
Technique: Substitute $x = y + 3$ into $5x + y = 21$
Check: Verify $5(4) + 1 = 21$ and $4 - 1 = 3$ ✓

Common Mistakes

Avoid these frequent errors

Adding equations before isolating a variable
Don't add $5x + y = 21$ and $x - y = 3$ immediately = you get $6x = 24$ and $x = 4$ , but then struggle to find y! This works here but isn't the substitution method. Always isolate one variable first, then substitute.

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

Why solve for x in the second equation instead of the first?

The second equation $x - y = 3$ is simpler to solve because the coefficient of x is 1. This avoids fractions and makes substitution easier!

What if I solved for y instead of x first?

That works too! You'd get $y = x - 3$ , then substitute into the first equation. You'll get the same answer: x = 4, y = 1.

How do I check if my solution is correct?

Substitute both values into both original equations:

$5(4) + 1 = 21$ ✓
$4 - 1 = 3$ ✓

If both check out, your solution is correct!

Can I use elimination instead of substitution?

Yes! For this system, you could add the equations directly since the y terms are opposites. Both methods give the same answer, so use whichever feels easier to you.

What does the solution (4, 1) mean graphically?

The point $(4, 1)$ is where the two lines intersect on a coordinate plane. It's the only point that satisfies both equations simultaneously!

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