Find the equation of a line joining the points (2, 2) and ( 2, 5).

To find the equation of a straight line passing through two given points $(x_1,y_1)$ and $(x_2,y_2)$, we use the two-point form of the equation of a line:

$$\frac{y-y_1}{y_2-y_1}=\frac{x-x_1}{x_2-x_1}$$

Given the points $(2,2)$ and $(-2,5)$, we substitute these values into the formula:

Step 1: Identify the coordinates

Let $(x_1,y_1)=(2,2)$ and $(x_2,y_2)=(-2,5)$.

Step 2: Calculate the slope ($m$)

The slope of the line is given by:

$$m=\frac{y_2-y_1}{x_2-x_1}=\frac{5-2}{-2-2}=\frac{3}{-4}=-\frac{3}{4}$$

Step 3: Use the point-slope form

The point-slope form of the equation of a line is:

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

Substituting $m=-\frac{3}{4}$, $x_1=2$, and $y_1=2$:

$$y-2=-\frac{3}{4}(x-2)$$

Step 4: Simplify to slope-intercept form ($y=mx+c$)

Multiply both sides by 4 to eliminate the fraction:

$$4(y-2)=-3(x-2)$$

$$4y-8=-3x+6$$

Bring all terms to one side:

$$3x+4y-8-6=0$$

$$3x+4y-14=0$$

Alternatively, we can express it in slope-intercept form:

$$4y=-3x+14$$

$$y=-\frac{3}{4}x+\frac{14}{4}$$

$$y=-\frac{3}{4}x+\frac{7}{2}$$

Final Equation

The equation of the line in general form is:

$3x+4y-14=0$

Alternatively, in slope-intercept form:

$y=-\frac{3}{4}x+\frac{7}{2}$