Question

How do you graph the system of equations `2x+y=6` and `6x+3y=18`?

Answer 1

See a solution process below

Explanation:

For each equation, we need to find and plot two points from the equation and then draw a line through the points to graph the systems of equations.

Equation 1

For `x = 0`

`(2 * 0) + y = 6`

`0 + y = 6`

`y = 6` or `(0, 6)`

For `x = 2`

`(2 * 2) + y = 6`

`4 + y = 6`

`-color(red)(4) + 4 + y = -color(red)(4) + 6`

`0 + y = 2`

`y = 2` or `(2, 2)`

We can now plot these two points and draw the line through them to graph the equation:

graph{(2x+y-6)(x^2+(y-6)^2-0.05)((x-2)^2+(y-2)^2-0.05)=0 [-15, 15, -7.5, 7.5]}

Equation 2

For `x = 0`

`(6 * 0) + 3y = 18`

`0 + 3y = 18`

`3y = 18`

`(3y)/color(red)(3) = 18/color(red)(3)`

`y = 6` or `(0, 6)`

For `x = 2`

`(6 * 2) + 3y = 18`

`12 + 3y = 18`

`-color(red)(12) + 12 + 3 = -color(red)(12) + 18`

`0 + 3y = 6`

`3y = 6`

`(3y)/color(red)(3) = 6/color(red)(3)`

`y = 2` or `(2, 2)`

As you can see the two points are the same, so the line is the same line as above. Both the equations in the problem represent the same line.

Answer 2

Line 1: (0, 6) and (3, 0)
Line 2: (0, 6) and (3, 0)
Then you connect the points.

Explanation:

The two lines are collinear, meaning that they are the same line. By dividing both sides of the second equation by 3, you can see that both lines are identical.

You can quickly graph any line with two points on the line. In this case, you assume that at some point, the line will cross the x-axis, making y zero. You can substitute 0 for y to find the corresponding x value. 2x + 0 = 6. x = 3. y=0 and x=3 yields the point (3, 0). Doing the same thing again, but setting x equal to zero, we find that y=6. This yields the point (0, 6). You can then plot these two points on a graph and draw a line with a straightedge that passes through both points.