Let's find the equation of the straight line and then determine the sum of the absolute values of its intercepts.

Finding the Point of Intersection

We are given two lines:

1. $x + 2y + 2 = 0 \quad \text{(1)}$
2. $2x - 3y - 3 = 0 \quad \text{(2)}$

To find the point of intersection, we can solve these two linear equations simultaneously.

From equation (1), we can express $x$ in terms of $y$:
$x = -2y - 2 \quad \text{(3)}$

Substitute equation (3) into equation (2):
$2(-2y - 2) - 3y - 3 = 0$
$-4y - 4 - 3y - 3 = 0$
$-7y - 7 = 0$
$-7y = 7$
$y = \frac{7}{-7} = -1$

Now substitute the value of $y = -1$ back into equation (3) to find $x$:
$x = -2(-1) - 2$
$x = 2 - 2$
$x = 0$

So, the point of intersection of the two lines is $(0, -1)$. The required straight line passes through this point.

Finding the Equation of the Straight Line

A straight line that cuts intercepts $a$ on the x-axis and $b$ on the y-axis has the equation in intercept form:

$\frac{x}{a} + \frac{y}{b} = 1 \quad \text{(4)}$

The problem states that the line cuts equal intercepts in the fourth quadrant.

• In the fourth quadrant, the x-coordinate is positive, and the y-coordinate is negative.
• Equal intercepts mean the absolute values of the intercepts are equal.
• So, if the x-intercept is $a$, the y-intercept is $b$, we must have $a > 0$, $b < 0$, and $|a| = |b|$. This implies $a = -b$.

Let the x-intercept be $k$ (where $k > 0$). Then the y-intercept must be $-k$. The equation of the line becomes:

$\frac{x}{k} + \frac{y}{-k} = 1$

Multiplying by $k$ (since $k \neq 0$):
$x - y = k \quad \text{(5)}$

Since the line passes through the point of intersection $(0, -1)$, this point must satisfy the line's equation (5). Substitute $x=0$ and $y=-1$ into equation (5):

$0 - (-1) = k$
$1 = k$

So, the value of $k$ is 1. This means the x-intercept is $a = k = 1$ and the y-intercept is $b = -k = -1$.

The intercepts are $a=1$ and $b=-1$. The x-intercept is positive, and the y-intercept is negative, confirming it cuts intercepts in the fourth quadrant. The absolute values are $|1|=1$ and $|-1|=1$, confirming they are equal intercepts (in magnitude).

Calculating the Sum of Absolute Values of Intercepts

The intercepts are $a = 1$ and $b = -1$.

The absolute value of the x-intercept is $|a| = |1| = 1$.

The absolute value of the y-intercept is $|b| = |-1| = 1$.

The sum of the absolute values of the intercepts is $|a| + |b| = 1 + 1 = 2$.

The straight line is $x - y = 1$. It passes through $(0, -1)$, and its intercepts are $1$ on the x-axis and $-1$ on the y-axis. The sum of the absolute values of the intercepts is $1 + 1 = 2$.