Estimate the solution to the system of these linear equations based on the graph

To estimate the solution to the system of equations given by:

1. \( y = -x + 6 \) (this is a straight line with a negative slope, crossing the y-axis at 6)
2. \( y = 3x - 2 \) (this is a straight line with a positive slope, crossing the y-axis at -2)

We need to find the point where these two lines intersect.

Finding the Intersection Algebraically

To find the exact intersection, we set the two equations equal to each other:

\[ -x + 6 = 3x - 2 \]

Next, we solve for \( x \):

1. Add \( x \) to both sides: \[ 6 = 3x + x - 2 \] \[ 6 = 4x - 2 \]

2. Add 2 to both sides: \[ 8 = 4x \]

3. Divide by 4: \[ x = 2 \]

Now that we have \( x \), we can substitute it back into either equation to find \( y \). We'll use the first equation:

\[ y = -x + 6 = -2 + 6 = 4 \]

Solution

Thus, the solution to the system of equations, which is the point of intersection, is:

\[ (x, y) = (2, 4) \]

Graphical Representation

If you were to graph these lines, you would see one line sloping downwards (from left to right) and the other sloping upwards, intersecting at the point (2, 4). This is where both equations are satisfied simultaneously.

Let's verify the solution step by step to ensure accuracy.

We have the two equations:

1. \( y = -x + 6 \)
2. \( y = 3x - 2 \)

Step 1: Set the equations equal

Since both equations equal \( y \), we can set them equal to each other:

\[ -x + 6 = 3x - 2 \]

Step 2: Solve for \( x \)

We will isolate \( x \) by first adding \( x \) to both sides:

\[ 6 = 3x + x - 2 \]

This simplifies to:

\[ 6 = 4x - 2 \]

Now, add \( 2 \) to both sides:

\[ 8 = 4x \]

Dividing both sides by \( 4 \):

\[ x = 2 \]

Step 3: Substitute \( x \) back to find \( y \)

Now, substitute \( x = 2 \) back into either original equation to find \( y \). We'll use the first equation:

\[ y = -x + 6 = -2 + 6 = 4 \]

Let's also check it with the second equation:

\[ y = 3x - 2 = 3(2) - 2 = 6 - 2 = 4 \]

Conclusion

Both equations yield the same value of \( y \) when \( x = 2 \).

Thus, the solution to the system of equations is:

\[ (x, y) = (2, 4) \]

So yes, the calculations are indeed accurate, and the solution is confirmed to be correct.