Question

1. What is an equation for the line with slope 2/3 and y-intercept 9?

Answer 1

Question 1:

 The generic equation of the line is given by:
 $y = mx + b$
 Where,
 m: slope of the line
 b: cutting point with the y axis
 Substituting values ​​we have:
 $y = \frac{2}3}x + 9$
 Answer:
 an equation for the line with slope 2/3 and y-intercept is:
 $y = \frac{2}3}x + 9$

 Question 2:

 The standard equation of the line is given by:
 $y-yo = m (x-xo)$
 Where,
 m: slope of the line
 (xo, yo): ordered pair that belongs to the line
 The slope of the line is:
 $m = \frac{y2-y1}{x2-x1}$
 Substituting values:
 $m = \frac{1-(-3)}{3-1}$
 $m = \frac{1+3}{3-1}$
 $m = \frac{4}{2}$
 $m = 2$
 We choose an ordered pair:
 $(xo, yo) = (3, 1)$
 Substituting values:
 $y-1 = 2 (x-3)$
 Rewriting:
 $y = 2x - 6 + 1 y = 2x - 5$
 Answer:
 An equation in slope-intercept form for the line that passes through the points (1, -3) and (3,1) is:
 $y = 2x - 5$

 Question 3:

 For this case, since the variation is direct, then we have an equation of the form:
 $y = kx$ Where,
 k: constant of variation.
 We then have the following equation:
 $-4y = 8x$ Rewriting we have:
 $y = \frac{8}{-4}x$
 $y = -2x$ Therefore, the constate of variation is given by:
 $k = -2$
 Answer:
 the constant of variation is:
 $k = -2$

 Question 4:

 For this case, since the variation is direct, then we have an equation of the form:
 $y = kx$ Where,
 k: constant of variation.
 We must find the constant k, for this we use the following data:
 y = 24 when x = 8
 Substituting values:
 $24 = k8$ Clearing k we have:
 $k = \frac{24}{8}$
 $k = 3$
 Therefore, the equation is:
 $y = 3x$ Thus, substituting x = 10 we have that the value of y is given by:
 $y = 3 (10) y = 30$ Answer:
 the value of y when x = 10 is:
 $y = 30$

Answer 2

(1) The equation of straight line for slope $\frac{2}{3}$ and y-intercept $9$ is $\boxed{y=\dfrac{2}{3}x+9}$.

(2) The equation of line that passes through the points $(1,-3)$ and $(3,1)$ is $\boxed{y=2x-5}$.

(3) The constant of variation for line $-4y=8x$ is $\boxed{k=-2}$.

(4) The value of $y$ is $\boxed{20}$.

Further explanation:

Part (1)

Concept used:

The equation of straight line for slope $m$ and intercept $c$ can be expressed as,

$\boxed{y=mx+c}$     ....(1)

Here, $c$ is the $y$-intercept of the straight line.

Given:

The slope of the line is $\frac{2}{3}$ and $y$-intercept is $9$.

Calculation:

Substitute $m=\frac{2}{3}$ and $c=9$ in the equation (1) to obtain the equation of straight line as follows,

$y=\dfrac{2}{3}x+9$  

Therefore, the equation of straight line for slope $\frac{2}{3}$ and $y$-intercept $9$ is $y=\frac{2}{3}x+9$.

Part (2)

Concept used:

The equation of straight line that passes through the point $(x_{1},y_{1})$ and $(x_{2},y_{2})$ is expressed as,

$\boxed{y-y_{1}=\dfrac{y_{2}-y_{1}}{x_{2}-x_{2}}(x-x_{1})}$  ....(2)

Here, $\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$ is the slope $(m)$ of the line.

Now, the equation of line in the point slope form is,

$\boxed{y-y_{1}=m(x-x_{1})}$  

Calculation:

The equation passes through the point $(1,-3)$ and $(3,1)$.

Now, substitute $1$ for $x_{1}$, $-3$ for $y_{1}$,$3$ for $x_{2}$ and $1$ for $y_{2}$ in the equation (2) to obtain the equation of line.

$\begin{aligned}y-(-3)&=\dfrac{1-(-3)}{3-1}(x-1)\\y+3&=\dfrac{4}{2}(x-1)\\y+3&=2(x-1)\\y&=2x-2-3\\y&=2x-5\end{aligned}$

Therefore, the equation in slope intercepts form for line that passes through the points $(1,-3)$ and $(3,1)$ is $y=2x-5$.

Part (3)

Given:

The equation of line is $-4y=8x$.

Concept used:

If the value of $y$ varies directly with the value of $x$ that means as the value of $x$ increases, $y$ increases in the same ratio.

$\boxed{y=kx}$

Here, $k$ is the constant of variation.

Calculation:

The given equation $-4y=8x$ can be simplified as,

$\begin{aligned}y&=-\dfrac{8x}{4}\\y&=-2x\end{aligned}$  

Compare the equation $y=-2x$ with the equation (3) to obtain the value of $k$.

$\boxed{k=-2}$  

Therefore, the constant of variation for the given equation is $k=-2$.

Part (4)

Calculation:

The value of $y$ is $24$ when $x$ is $8$. The value of $y$ varies directly with $x$.

The constant of variation $k$ is calculated as,

$\begin{aligned}k&=\dfrac{24}{8}\\&=3\end{aligned}$  

The constant of variation $k$ is $3$.

The relationship between $x$ and $y$ can be written as,

$\boxed{y=kx}$      ......(4)

Substitute $10$ for $x$ and $3$ for $k$ in the equation (4) to obtain the value of $y$.

$\begin{aligned}y&=3\times10\\&=30\end{aligned}$  

Therefore, the value of $y$ is $30$.

Learn more:

1. A problem on line brainly.com/question/1473992

2. A problem on simplification brainly.com/question/3658196

3. A problem on permutation brainly.com/question/5199020

Answer details:

Grade: Middle school

Subject: Mathematics

Chapter: Coordinate geometry

Keywords: Slope, equation of line, constant of variation, y=2/3x+9, points,  intercepts, straight line, slope-intercept form.