Question 559100
{{{3y+6=2x}}} is a linear equation. Its graph is a straight line.
To graph it you just need to find and plot two points, and then connect them with a straight line. Chosing the points is an art, and practice improves your artistic ability. Making {{{x=0}}} and solving for {{{y}}} is often a good strategy. Making {{{y=0}}} and solving for {{{x}}} could work too.
For your equation:
Making {{{x=0}}} results in {{{3y+6=0}}} --> {{{3y=-6}}} --> {{{y=-2}}} , giving you the pioint (0,-2) .
Making {{{y=0}}} results in {{{0+6=2x}}} --> {{{2x=6}}} --> {{{x=3}}} , giving you the point (3,0).
With those two points I would draw the x-axis from -2 to 5 (to include {{{x=0}}} , and {{{x=3}}} , and a little beyond on both sides). I would draw the y-axis from -4 to 2 for similar reasons. You could use wider ranges on both axes too.
The slope and y-intercept can be seen from the graph, or can be "read" from the slope-intercept equation for the line.
In the graph, you see the point where the line crosses (intercepts) the y-axis. It's (0,-2). The x-coordinate of the point where the line crosses the y- axis is always zero, of course. The y-coordinate is what we want to know, and that value is what we call the y-intercept.
You can also see from the graph that to go from (0,-2) to (3,0) you run 3 spaces to the right and rise 2 spaces straight up. The slope is the ratio of that rise, divided by that run, or {{{2/3}}}. You can calculate it from any 2 points in the graph. For a straight line, it will always be the same, whatever points you chose.
There are infinite ways to write the equation for a line. You were giveen
{{{3y+6=2x}}} , but the same line could be described by many equivalent equations:
{{{3y+6=2x}} --> {{{2x-3y-6=0}}} --> {{{2x-3y=6}}} --> {{{4x-6y=12}}} --> {{{20x-30y=60}}} and infinite more.
My favorite version is {{{y=(2/3)x-2}}} . That is called the slope-intercept form. There are many ways to write the equation for a line, but there is only one slope-intercept form for each line. To find it, you just solve for {{{y}}}.
In the slope-intercept form of the equation for the line, the number that multiplies the {{{x}}} (with a minus sign if there is one) is the slope, and the other term (with a minus sign if there is one) is the y-intercept. In
{{{y=(2/3)x-2}}} you see that the slope is {{{2/3}}} and the y-intercept is {{{-2}}}, even if you do not graph the line.