Solve the Linear System: Find Solutions for x/2 + y = 3, x + 2y = 6

Q: What does it mean when I get 6 = 6 after substitution?

When you get a true statement like 6 = 6, it means both equations describe the same line! Every point on that line is a solution, giving you infinite solutions.

Q: How is this different from getting 6 = 8?

If you got 6 = 8 (a false statement), the system would have no solution because the lines are parallel. But 6 = 6 is always true, meaning the lines overlap completely.

Q: Can I write out some specific solutions?

Yes! Use $ x = 6 - 2y $. Pick any y-value: if y = 0, then x = 6. If y = 1, then x = 4. If y = 2, then x = 2. All of these work!

Q: How do I recognize dependent equations earlier?

Check if one equation is a multiple of the otherMultiply the first equation by 2: $ x + 2y = 6 $This matches the second equation exactly!

Linear Systems with Dependent Equations

$\frac{x}{2}+y=3$

$x+2y=6$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:03 Let's multiply one of the equations by 2, so we can subtract between them

00:11 Now let's subtract between the equations

00:14 Let's simplify what we can

00:20 There are infinite solutions

00:27 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

$\frac{x}{2}+y=3$

$x+2y=6$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Express one equation in terms of a single variable.
Step 2: Substitute into the other equation.
Step 3: Evaluate and determine the solution type.

Now, let's work through each step:
Step 1: Solve the first equation for $x$ :
$\frac{x}{2} + y = 3 \quad \Rightarrow \quad x = 6 - 2y$ .

Step 2: Substitute $x = 6 - 2y$ into the second equation:
$x + 2y = 6 \quad \Rightarrow \quad (6 - 2y) + 2y = 6$ .

Step 3: Simplifying equation:
$6 - 2y + 2y = 6 \quad \Rightarrow \quad 6 = 6$ .

This equation $6 = 6$ is always true, indicating that both equations are dependent and represent the same line.

Therefore, the system has infinite solutions.

Final Answer

Infinite solutions

Key Points to Remember

Essential concepts to master this topic

Substitution: Solve first equation for one variable completely
Technique: From $\frac{x}{2} + y = 3$ , get $x = 6 - 2y$
Check: When substitution gives 6 = 6, system has infinite solutions ✓

Common Mistakes

Avoid these frequent errors

Assuming one specific solution exists when getting a true statement
Don't stop at x = 2, y = 2 when you get 6 = 6 = wrong conclusion! The true statement means every point on the line satisfies both equations. Always recognize that 6 = 6 indicates infinite solutions, not one specific answer.

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

What does it mean when I get 6 = 6 after substitution?

When you get a true statement like 6 = 6, it means both equations describe the same line! Every point on that line is a solution, giving you infinite solutions.

How is this different from getting 6 = 8?

If you got 6 = 8 (a false statement), the system would have no solution because the lines are parallel. But 6 = 6 is always true, meaning the lines overlap completely.

Can I write out some specific solutions?

Yes! Use $x = 6 - 2y$ . Pick any y-value: if y = 0, then x = 6. If y = 1, then x = 4. If y = 2, then x = 2. All of these work!

Why can't I just solve normally and get one answer?

You tried to solve normally! But when the equations represent the same line, there isn't just one intersection point - the entire line is the intersection.

How do I recognize dependent equations earlier?

Check if one equation is a multiple of the other
Multiply the first equation by 2: $x + 2y = 6$
This matches the second equation exactly!

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