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  1. What is an equation for the line with slope 2/3 and y-intercept 9?
  2. What is an equation in slope-intercept form for the line that passes through the points (1,-3) and (3,1).
  3. For the equation -4y = 8x, what is the constant of variation?
  4. Suppose y varies directly with x, and y = 24 when x=8. What is the value of y when x = 10?

Verified Answer :

carlosego

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:
 \(y = 30\)

(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 https://brainly.com/question/1473992

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

3. A problem on permutation https://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.