The graph of a line in the xy-plane has slope `(1)/(2)` and contains the point `(0, 7)`. The graph of a seconds line passes through the points `(0, 0) and (-1, 3)`. If the two lines intersect at the point `(r, s)`, what is the value of `r+s`?

Options

A. `-3`
B. `-2`
C. `2`
D. `4`

Answer

To solve the problem step by step, we will find the equations of both lines and then determine their intersection point.

### Step 1: Find the equation of the first line
The first line has a slope of \( \frac{1}{2} \) and passes through the point \( (0, 7) \). The equation of a line in slope-intercept form is given by:

\[ y = mx + c \]

Where \( m \) is the slope and \( c \) is the y-intercept. Here, \( m = \frac{1}{2} \) and \( c = 7 \).

So, the equation of the first line is:

\[ y = \frac{1}{2}x + 7 \]

### Step 2: Find the equation of the second line
The second line passes through the points \( (0, 0) \) and \( (-1, 3) \). We can use the two-point form of the equation of a line:

\[ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) \]

Let \( (x_1, y_1) = (0, 0) \) and \( (x_2, y_2) = (-1, 3) \).

Substituting these values into the formula:

\[ y - 0 = \frac{3 - 0}{-1 - 0}(x - 0) \]

This simplifies to:

\[ y = -3x \]

### Step 3: Set the equations equal to find the intersection point
Now we have the equations of both lines:

1. \( y = \frac{1}{2}x + 7 \)
2. \( y = -3x \)

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

\[ \frac{1}{2}x + 7 = -3x \]

### Step 4: Solve for \( x \)
Rearranging the equation:

\[ \frac{1}{2}x + 3x = -7 \]

Combining like terms:

\[ \frac{1}{2}x + \frac{6}{2}x = -7 \]

\[ \frac{7}{2}x = -7 \]

Multiplying both sides by \( \frac{2}{7} \):

\[ x = -2 \]

### Step 5: Find \( y \) using one of the equations
Now we can substitute \( x = -2 \) into either equation to find \( y \). Using the second equation \( y = -3x \):

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

### Step 6: Determine the intersection point
The intersection point is \( (-2, 6) \).

### Step 7: Calculate \( r + s \)
Here, \( r = -2 \) and \( s = 6 \). Therefore:

\[ r + s = -2 + 6 = 4 \]

### Final Answer
The value of \( r + s \) is \( 4 \).

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