Question

Find the equations of the line that passes trough point (3,4)and is parallel to the given line. 3x+2y=4

Answer 1

It is required that you find the equation of a line that passes through the point (3,4) and is parallel to 3x+2y=4.

Recall that the equation of a line with slope m and passes through the point (x₁,y₁) using the point-slope form is given as:

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

This implies that what is left to find the equation of the required line is its slope.

Recall that the Slopes of parallel lines are equal (the same).

Write the given equation 3x+2y=4 in slope-intercept form, y=mx+c, as follows:

`\begin{gathered} 3x+2y=4 \\ \Rightarrow2y=-3x+4 \\ \Rightarrow y=-(3)/(2)x+(4)/(2) \\ \Rightarrow y=-(3)/(2)x+2 \end{gathered}`

Compare this equation with the slope-intercept form, y=mx+c, it follows that the slope of the given line is m=-3/2.

Since the slopes of parallel lines are equal, it follows that the slope of the required line is -3/2.

Substitute the point (x₁,y₁)=(3,4) and m=-3/2 into the formula for the equation of a line in the point-slope form:

`\begin{gathered} y-4=-(3)/(2)(x-3) \\ \Rightarrow y-4=-(3x)/(2)-(3)/(2)(-3) \\ \Rightarrow y-4=-(3x)/(2)+(9)/(2) \\ \Rightarrow y=-(3x)/(2)+(9)/(2)+4 \\ \Rightarrow y=-(3)/(2)x+(17)/(2) \end{gathered}`

The required equation is y= -3/2 x + 17/2.