Question

Find the answers for the given question? Slope-intercept? Point slope? Standard?

Answer 1

`\begin{gathered} \text{Slope}-\text{intercept: }y=-(1)/(2)x+1 \\ \text{ Point-slope form: y-2=-}(1)/(2)(x+2) \\ \text{ Standard form: x+2y=2} \end{gathered}`

Explanations:

The given functions can be generalized using the form f(x) = y

Given the following functions;

`\begin{gathered} f(-2)=2 \\ f(8)=-3 \end{gathered}`

These functions can be written as coordinates points (-2, 2) and (8, -3)

The equation of the linear function in slope-intercept form is expressed as y = mx + b

m is the slope:

b is the y-intercept

Get the slope of the line passing through the points:

`\begin{gathered} m=(y_2-y_1)/(x_2-x_1) \\ m=(-3-2)/(8-(-2)) \\ m=(-5)/(8+2) \\ m=-(5)/(10) \\ m=-(1)/(2) \end{gathered}`

Get the y-intercept using the point(-2, 2) and m = -0.5

`\begin{gathered} y=mx+b \\ 2=-0.5(-2)+b \\ 2=1+b \\ b=2-1 \\ b=1 \end{gathered}`

Write the equation in slope-intercept form where m = -0.5 and b = 1;

`y=-(1)/(2)x+1`

Write in point-slope form;

The point-slope form of the equation is expressed as;

`y-y_1=m(x-x_1)`

Using the following parameters;

`\begin{gathered} m=-(1)/(2) \\ (x_1,y_1)=(-2,2) \end{gathered}`

Substitute the given parameters into the point-slope form of the equation;

`\begin{gathered} y-2=-(1)/(2)(x-(-2)_{}) \\ y-2=-(1)/(2)(x+2) \end{gathered}`

This gives the point-slope form of the equation.

For the standard form:

The standard form of the linear equation is expressed as:

`Ax+By=C`

Recall that;

`y=-(1)/(2)x+1`

Rearrange in standard form as shown:

`\begin{gathered} 2y=-x+2 \\ \end{gathered}`

Add "x" to both sides of the equation:

`\begin{gathered} 2y+x=-x+x+2 \\ 2y+x=2 \\ x+2y=2 \end{gathered}`

This gives the required linear equation in standard form.