Question

A coordinate plane with a line passing through points (1, 2) and (4, 4) The slope of the graphed line is . Which formulas represent the line that is graphed? Check all that apply. y – 1 = (x – 2) y – 2 = (x – 1) y – 4 = (x – 4) f(x) = x + f(x) = x +

Answer 1

A straight line is represented by a linear equation.

The possible equations are:

\mathbf{3y = 2x + 4}3y=2x+4 .

\mathbf{y = \frac 23x + \frac{4}{3}}y=

3

2

x+

3

4

.

\mathbf{y = \frac 23(x - 1) + 2}y=

3

2

(x−1)+2 .

\mathbf{y = \frac 23(x - 4) + 4}y=

3

2

(x−4)+4

The points are given as:

\mathbf{(x,y) = \{(1,2),(4,4)\}}(x,y)={(1,2),(4,4)}

Start by calculating the slope (m)

\mathbf{m = \frac{y_2 - y_1}{x_2 - x_1}}m=

x

2

−x

1

y

2

−y

1

So, we have:

\mathbf{m = \frac{4-2}{4-1}}m=

4−1

4−2

\mathbf{m = \frac{2}{3}}m=

3

2

The equation in slope intercept form is then calculated as:

\mathbf{y = m(x - x_1) + y_1}y=m(x−x

1

)+y

1

So, we have:

\mathbf{y = \frac 23(x - 1) + 2}y=

3

2

(x−1)+2

or

\mathbf{y = \frac 23(x - 4) + 4}y=

3

2

(x−4)+4

Expand \mathbf{y = \frac 23(x - 1) + 2}y=

3

2

(x−1)+2

\mathbf{y = \frac 23x - \frac 23 + 2}y=

3

2

x−

3

2

+2

Take LCM

\mathbf{y = \frac 23x + \frac{-2 + 6}{3}}y=

3

2

x+

3

−2+6

\mathbf{y = \frac 23x + \frac{4}{3}}y=

3

2

x+

3

4

Multiply through by 3

\mathbf{3y = 2x + 4}3y=2x+4

So, the possible equations are:

\mathbf{3y = 2x + 4}3y=2x+4 .

\mathbf{y = \frac 23x + \frac{4}{3}}y=

3

2

x+

3

4

.

\mathbf{y = \frac 23(x - 1) + 2}y=

3

2

(x−1)+2 .

\mathbf{y = \frac 23(x - 4) + 4}y=

3

2

(x−4)+4