Question

18 multiple answer 4 points under which of the following conditions must two lines intersect in exactly one point? both lines have the same y - intercept. both lines have different slopes but the same x - intercept. both lines have the same y - intercept but different slopes. both lines have the same slope and the same x - intercept. 19 multiple choice 4 points what is the y - intercept of the line 3x + 4y = 12? (4,0) (0,4) (3,0) (0,3)

Explanation:

18.

Step1: Recall line - intersection concept

Two non - parallel lines (lines with different slopes) intersect at exactly one point.

Step2: Analyze each option

• Option 1: Same y - intercept doesn't guarantee single intersection. If slopes are equal, they are parallel or the same line.
• Option 2: Different slopes and same x - intercept means they intersect at that x - intercept point only.
• Option 3: Different slopes and same y - intercept means they intersect at the y - intercept point only.
• Option 4: Same slope and same x - intercept means they are the same line (infinite intersection points).

Step1: Rewrite the line equation in slope - intercept form

The slope - intercept form of a line is \(y=mx + b\), where \(b\) is the y - intercept. Given \(3x + 4y=12\), solve for \(y\).
\[

$$\begin{align*} 4y&=-3x + 12\\ y&=-\frac{3}{4}x+3 \end{align*}$$

\]
The y - intercept occurs when \(x = 0\). Substituting \(x = 0\) into the original equation \(3x + 4y=12\), we get \(4y=12\), so \(y = 3\). The y - intercept is the point \((0,3)\).

Answer:

B. Both lines have different slopes but the same x - intercept.
C. Both lines have the same y - intercept but different slopes.

19.