Question

How do you graph \[y = \left( {\dfrac{2}{3}} \right)x - 3?\]

Answer 1

Hint: Given a problem to first identify the kind of given equation. Then find out the points of the given equation. So apply a few \[x\] values to find a few \[y\] values on the given equation. Now we have points of the equation. In these points, we arrange them correctly. Now labeled those points in the graph. Finally, we get the graph of the straight line.


Complete step-by-step answer:
Given equation is a linear equation and this equation is the straight-line format. It means this is a straight line equation. The standard form linear equation is \[ax + by = c\] here, \[a,b\] and \[c\] are some constant values. And if a linear equation is in \[y = mx + c\] format, then the equation has a slope and intercept value. So the given equation has a slope and an intercept.
The linear equation is \[y = \left( {\dfrac{2}{3}} \right)x - 3\]
Any line equation can be graphed using some points, select few \[x\] values and plug them into the equation. then we find the corresponding \[y\] values. So we assume \[x\] values to find \[y\] values.
Assume that,
 \[x = - 4\] apply the \[x\] value in equation \[y\] ,
 \[
  y = \left( {\dfrac{2}{3}} \right)( - 4) - 3 \\
   = - \dfrac{8}{3} - 3 \\
   = \dfrac{{ - 8 - 9}}{3} \\
   = \dfrac{{ - 17}}{3} \\
 \]
 \[x = - 3\] apply the \[x\] value in equation \[y\] ,
  \[
  y = \left( {\dfrac{2}{3}} \right)( - 3) - 3 \\
   = ( - 2) - 3 \\
   = - 2 - 3 \\
   = - 5 \\
 \]
 \[x = - 2\] apply the \[x\] value in equation \[y\] ,
 \[
  y = \left( {\dfrac{2}{3}} \right)( - 2) - 3 \\
   = (\dfrac{{ - 4}}{3}) - 3 \\
   = \dfrac{{ - 4 - 9}}{3} \\
   = \dfrac{{ - 13}}{3} \\
 \]
 \[x = - 1\] apply the \[x\] value in equation \[y\] ,
  \[
  y = \left( {\dfrac{2}{3}} \right)( - 1) - 3 \\
   = (\dfrac{{ - 2}}{3}) - 3 \\
   = \dfrac{{ - 2 - 9}}{3} \\
   = \dfrac{{ - 11}}{3} \\
 \]
 \[x = 0\] apply the \[x\] value in equation \[y\] ,
 \[
  y = \left( {\dfrac{2}{3}} \right)(0) - 3 \\
   = (0) - 3 \\
   = - 3 \\
 \]
 \[x = 1\] apply the \[x\] value in equation \[y\] ,
 \[
  y = \left( {\dfrac{2}{3}} \right)(1) - 3 \\
   = (\dfrac{2}{3}) - 3 \\
   = \dfrac{{2 - 9}}{3} \\
   = \dfrac{{ - 7}}{3} \\
 \]
 \[x = 2\] apply the \[x\] value in equation \[y\] ,
 \[
  y = \left( {\dfrac{2}{3}} \right)(2) - 3 \\
   = (\dfrac{4}{3}) - 3 \\
   = \dfrac{{4 - 9}}{3} \\
   = \dfrac{{ - 5}}{3} \\
 \]
 \[x = 3\] apply the \[x\] value in equation \[y\] ,
 \[
  y = \left( {\dfrac{2}{3}} \right)(3) - 3 \\
   = (2) - 3 \\
   = - 1 \\
 \]
 \[x = 4\] apply the \[x\] value in equation \[y\] ,
 \[
  y = \left( {\dfrac{2}{3}} \right)(4) - 3 \\
   = (\dfrac{8}{3}) - 3 \\
   = \dfrac{{8 - 9}}{3} \\
   = \dfrac{{ - 1}}{3} \\
 \]
Points are, \[( - 4, - \dfrac{{17}}{3}),( - 3, - 5),( - 2, - \dfrac{{13}}{3}),( - 1, - \dfrac{{11}}{3}),(0, - 3),(1, - \dfrac{7}{3}),(2,\dfrac{{ - 5}}{3}),(3, - 1),(4,\dfrac{{ - 1}}{3})\]
Label the points in the graph and draw a straight line.
Then we get a graph of \[y = 4x - 5\]
The graph is

Note: In linear equation is given the straight-line graph. Find points of the equation in some particular interval. So given the problem, find the values in between t \[x \in \left[ { - 4,4} \right]\] this interval. Now find the values in a small region. Find the points labeled to the graph. Now we get the straight-line graph on the given equation. And carefully, labeled the find out points in the graph.