QuestionFind the $x$ - and $y$ -intercepts of the following lines.
1. $2 x+3 y=24$
2. $3 x-5 y=30$
3. $7 x-4 y=84$
4. $x+y=8$ $5, x=5$
6. $3 y=66$

Studdy Solution

STEP 1

1. The $x$ -intercept of a line is the point where the line crosses the $x$ -axis, which occurs when $y = 0$ .
2. The $y$ -intercept of a line is the point where the line crosses the $y$ -axis, which occurs when $x = 0$ .
3. Each line equation can be solved separately for its intercepts.

STEP 2

1. Find the $x$ -intercept for each line.
2. Find the $y$ -intercept for each line.

STEP 3

For the line $2x + 3y = 24$ , find the $x$ -intercept by setting $y = 0$ :
$2x + 3(0) = 24$ $2x = 24$ $x = \frac{24}{2} = 12$
The $x$ -intercept is $(12, 0)$ .

STEP 4

For the line $2x + 3y = 24$ , find the $y$ -intercept by setting $x = 0$ :
$2(0) + 3y = 24$ $3y = 24$ $y = \frac{24}{3} = 8$
The $y$ -intercept is $(0, 8)$ .

STEP 5

For the line $3x - 5y = 30$ , find the $x$ -intercept by setting $y = 0$ :
$3x - 5(0) = 30$ $3x = 30$ $x = \frac{30}{3} = 10$
The $x$ -intercept is $(10, 0)$ .

STEP 6

For the line $3x - 5y = 30$ , find the $y$ -intercept by setting $x = 0$ :
$3(0) - 5y = 30$ $-5y = 30$ $y = \frac{30}{-5} = -6$
The $y$ -intercept is $(0, -6)$ .

STEP 7

For the line $7x - 4y = 84$ , find the $x$ -intercept by setting $y = 0$ :
$7x - 4(0) = 84$ $7x = 84$ $x = \frac{84}{7} = 12$
The $x$ -intercept is $(12, 0)$ .

STEP 8

For the line $7x - 4y = 84$ , find the $y$ -intercept by setting $x = 0$ :
$7(0) - 4y = 84$ $-4y = 84$ $y = \frac{84}{-4} = -21$
The $y$ -intercept is $(0, -21)$ .

STEP 9

For the line $x + y = 8$ , find the $x$ -intercept by setting $y = 0$ :
$x + 0 = 8$ $x = 8$
The $x$ -intercept is $(8, 0)$ .

STEP 10

For the line $x + y = 8$ , find the $y$ -intercept by setting $x = 0$ :
$0 + y = 8$ $y = 8$
The $y$ -intercept is $(0, 8)$ .

STEP 11

For the line $x = 5$ , find the $x$ -intercept. Since $x$ is always 5, the line is vertical and does not cross the $x$ -axis at any other point. Thus, there is no $x$ -intercept other than the point itself, which is $(5, 0)$ .

STEP 12

For the line $x = 5$ , find the $y$ -intercept. Since the line is vertical, it does not cross the $y$ -axis. Therefore, there is no $y$ -intercept.

STEP 13

For the line $3y = 66$ , find the $x$ -intercept. Since $y$ is always a constant value, the line is horizontal and does not cross the $x$ -axis. Therefore, there is no $x$ -intercept.

STEP 14

For the line $3y = 66$ , find the $y$ -intercept by setting $x = 0$ :
$3y = 66$ $y = \frac{66}{3} = 22$
The $y$ -intercept is $(0, 22)$ .

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