Draw the graph for each of the equations `x+y=6 and x-y=2` on the same graph paper and find the coordinates of the point where the two straight lines intersect.

Answer

To solve the problem, we will follow these steps:

### Step 1: Rewrite the equations in slope-intercept form
We have two equations:
1. \( x + y = 6 \)
2. \( x - y = 2 \)

We will rewrite both equations in the form \( y = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept.

**For the first equation:**
\[ x + y = 6 \]
Subtract \( x \) from both sides:
\[ y = 6 - x \]
This can be rewritten as:
\[ y = -x + 6 \]

**For the second equation:**
\[ x - y = 2 \]
Subtract \( x \) from both sides:
\[ -y = 2 - x \]
Multiply through by -1:
\[ y = x - 2 \]

### Step 2: Find the points to plot for each equation
We will find at least two points for each line to plot them on the graph.

**For the first equation \( y = -x + 6 \):**
- If \( x = 0 \):
\[ y = -0 + 6 = 6 \]
Point: \( (0, 6) \)

- If \( x = 6 \):
\[ y = -6 + 6 = 0 \]
Point: \( (6, 0) \)

**For the second equation \( y = x - 2 \):**
- If \( x = 0 \):
\[ y = 0 - 2 = -2 \]
Point: \( (0, -2) \] - If \( x = 4 \):
\[ y = 4 - 2 = 2 \]
Point: \( (4, 2) \] ### Step 3: Plot the points on graph paper 1. Plot the points \( (0, 6) \) and \( (6, 0) \) for the first equation.
2. Plot the points \( (0, -2) \) and \( (4, 2) \) for the second equation.

### Step 4: Draw the lines
- Draw a straight line through the points of the first equation.
- Draw a straight line through the points of the second equation.

### Step 5: Find the intersection point
To find the intersection point, we can either look at the graph where the two lines cross or solve the equations simultaneously.

**Solving the equations:**
We have:
1. \( x + y = 6 \) (Equation 1)
2. \( x - y = 2 \) (Equation 2)

We can add the two equations:
\[ (x + y) + (x - y) = 6 + 2 \]
This simplifies to:
\[ 2x = 8 \implies x = 4 \]

Now substitute \( x = 4 \) back into Equation 1:
\[ 4 + y = 6 \implies y = 2 \]

Thus, the intersection point is \( (4, 2) \).

### Final Answer
The coordinates of the point where the two straight lines intersect are \( (4, 2) \).

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