Question

What is the y-intercept for this linear equation \[\dfrac{1}{2}x - \dfrac{2}{3}y = - 6\] ?

Answer 1

Hint: To figure out the value of the y-intercept, transform the given equation into its proper form, and then compare the values of the general equation with the proper form of the equation.

Complete step-by-step solution:
The y-intercept, or the \[Y - axis\] intercept of a straight line equation is the point where the line cuts the \[Y - axis\] axis on the Cartesian graph. The coordinate of the point of intersection of \[Y - axis\] and the straight line is called the \[Y - axis\] intercept, and \[c\] is used to denote it in the general equation, \[y = mx + c\]. In order to find out the y-intercept, we will have to compare this equation with the general form of the equation and find out the value of \[c\] by comparing values. However, the equation given to us in the question, as it stands, contains fractions and thus is not in the proper form. To convert this equation into the ideal form, we will have to multiply this equation with the LCM (Least Common Multiple) of all the terms in the denominator. In this equation, there are two fraction coefficients involved, and the LCM of the denominator values $2$ and $3$ is, \[2 \times 3 = 6\]. Hence, we will multiply the given equation with $6$ on the LHS and RHS to get rid of the fraction coefficients and convert the equation into its proper form.
\[\dfrac{1}{2}x - \dfrac{2}{3}y = - 6\]
Multiplying equation with 6, we get
\[(\dfrac{1}{2}x - \dfrac{2}{3}y = - 6) \times 6\] = \[3x - 4y = - 36\]
Hence the equation in its proper form is,
\[3x - 4y = - 36\] ……………………...\[(1.1)\]
Now, to compare this equation with the general equation\[y = mx + c\], we need to transform this equation into a similar form. Thus,
\[4y = 3x + 36\] ……………………….\[(1.2)\]
Now dividing equation\[(1.2)\]by $4$ on both the LHS and RHS we get,
\[y = \dfrac{3}{4}x + 9\] ………………………..\[(1.3)\]
Now, comparing this equation with the general form \[y = mx + c\] we can see that,
\[y = mx + c\]
\[y = \dfrac{3}{4}x + 9\]
Hence, on comparing the values of \[m\] and \[c\], we can see that,
\[m\]= \[\dfrac{3}{4}\]
\[c\]= \[9\]
Thus, the value of the \[Y - axis\] intercept for the equation \[\dfrac{1}{2}x - \dfrac{2}{3}y = - 6\] is $9$.
Note: Keep in mind that to find out the value of slope or Y-intercept of a straight line equation, it should be in proper form, so that it can be compared with the general form of the straight line equation.