Coordinate Geometry - The y-intercept-advanced

Find the $y$-intercept of the line given by the equation $3x - 5y + 15 = 0$. Express the answer as a coordinate.

Given equation: $3x - 5y + 15 = 0$. To find the $y$-intercept, we set $x = 0$: $3(0) - 5y + 15 = 0$ $-5y + 15 = 0$ $-5y = -15$ $y = \frac{-15}{-5}$ $y = 3$ Therefore, the $y$-intercept is $(0, 3)$.

At the $y$-intercept, the value of $x$ is zero. By substituting $x=0$ into the linear equation, we isolate the $y$ variable to find its specific value on the $y$-axis.

A line passes through the points $A(2, 5)$ and $B(4, 9)$. Determine the $y$-intercept of this line.

Step 1: Find the slope ($m$) using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$: $m = \frac{9 - 5}{4 - 2} = \frac{4}{2} = 2$ Step 2: Use the slope-intercept form $y = mx + c$ and substitute point $A(2, 5)$: $5 = 2(2) + c$ $5 = 4 + c$ $c = 5 - 4$ $c = 1$ The $y$-intercept is $1$, or the point $(0, 1)$.

First, the slope is calculated. Then, using one of the given points and the slope, we solve for the constant $c$ in $y=mx+c$, which represents the $y$-intercept.

Find the area of the triangle formed by the line $4x + 3y = 24$ and the coordinate axes.

Step 1: Find the $x$-intercept (set $y = 0$): $4x + 3(0) = 24 \implies 4x = 24 \implies x = 6$ The $x$-intercept is at $A(6, 0)$. Step 2: Find the $y$-intercept (set $x = 0$): $4(0) + 3y = 24 \implies 3y = 24 \implies y = 8$ The $y$-intercept is at $B(0, 8)$. Step 3: Calculate area of triangle $OAB$ where $O$ is $(0, 0)$: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ $\text{Area} = \frac{1}{2} \times 6 \times 8$ $\text{Area} = 24 \text{ sq. units}$

The $x$ and $y$ intercepts provide the base and height of the right-angled triangle formed with the origin. The $y$-intercept provides the vertical height from the origin.

1. From the graph, the line crosses the $y$-axis at the point $(0, -4)$. Therefore, the $y$-intercept is $-4$.
2. To find the slope ($m$), use the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ with points $(0, -4)$ and $(2, 0)$.
3. $m = \frac{0 - (-4)}{2 - 0} = \frac{4}{2} = 2$.
4. Thus, the $y$-intercept is $-4$ and the slope is $2$.

The $y$-intercept is the $y$-coordinate where $x=0$. Visually, this is the point where the line 'hits' the vertical axis.

1. Parallel lines have equal slopes. The slope of $y = -3x + 10$ is $-3$.
2. The given $y$-intercept is $5$, so $c = 5$.
3. Using the slope-intercept form $y = mx + c$, the equation is $y = -3x + 5$.
4. The line passes through $(0, 5)$ and has a downward slope.

Since the lines are parallel, we inherit the slope. The $y$-intercept provides the specific vertical starting point for our new line.