Exercise 49 Page 429

We will start by writing functions that represent the lines $\ell$ and $p.$ Then, to find the intersection point, we will solve the functions simultaneously.

Writing Function for Line $\ell$

We will start by finding the slope of the line $\ell .$ To do so, we will use the points $(0,\text{-}3)$ and $(2,7).$ Let's name the points. To calculate the slope, we will substitute the points into the Slope Formula. The slope of the line $\ell$ is $5.$ We can write the function for the line $\ell$ using the slope-intercept form because we know its slope $5$ and $y\text{-}$intercept $(0,\text{-}3).$

Writing Function for Line $p$

Since the line $p$ is perpendicular to line $\ell ,$ the slope of the line $p$ is opposite reciprocal of $5,$ $\text{-}\frac{1}{5}.$ We know the slope of the line $p$ and a point $(3,\text{-}1)$ on it. Hence, we can write the linear function that represents the line $p$ using the slope-intercept form. Let's substitute the point $(3,\text{-}1)$ in this form to find the value of $b.$ Therefore, the function for line $p$ is $y=\text{-}\frac{1}{5}x-\frac{2}{5}.$

Solving System of Equations

Now, to find the point of intersection, we need to solve the system of equations. We will use the Substitution Method to solve the system. The $x\text{-}$coordinate of the intersection is $\frac{1}{2}.$ The answer is A.