The lines with equations 3𝑦+𝑥+1=0
 and 5𝑦+2𝑥+2=0
 intersect at the point (𝑎,𝑏)
. What are the values of 𝑎
 and 𝑏
?

To find the point of intersection \((a, b)\) of the two lines represented by the equations:

1. \( 3y + x + 1 = 0 \)
2. \( 5y + 2x + 2 = 0 \)

We need to solve this system of linear equations for \(x\) and \(y\).

### Step 1: Express one variable in terms of the other
Let's solve the first equation for \(x\) in terms of \(y\):

\[
3y + x + 1 = 0 \implies x = -3y - 1
\]

### Step 2: Substitute this into the second equation
Now, substitute \(x = -3y - 1\) into the second equation:

\[
5y + 2(-3y - 1) + 2 = 0
\]

Simplify the equation:

\[
5y - 6y - 2 + 2 = 0
\]
\[
-y = 0
\]
\[
y = 0
\]

### Step 3: Find the corresponding \(x\)
Substitute \(y = 0\) into \(x = -3y - 1\):

\[
x = -3(0) - 1 = -1
\]

### Conclusion
The point of intersection is \((a, b) = (-1, 0)\).

Do you want more details or have any questions?

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Here are 5 related questions for further exploration:

1. How can you solve a system of linear equations graphically?
2. What happens if the two lines were parallel? How would that affect the solution?
3. How can substitution and elimination methods compare when solving such systems?
4. How do you interpret the slope and y-intercept of these lines?
5. Can you solve this system using matrix methods?

**Tip:** Always check your solutions by plugging the values back into the original equations!

The lines with equations 3𝑦 + 𝑥 + 1 = 0 and 5𝑦 + 2𝑥 + 2 = 0 intersect at the point (𝑎,𝑏). What are the values of 𝑎 and 𝑏?

Learn how to find the point of intersection between two linear equations, 3y + x + 1 = 0 and 5y + 2x + 2 = 0, using substitution. This solution covers algebraic concepts related to systems of equations and linear intersections. Suitable for students in Grades 9-11.

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