a In this exercise our plan is to sketch the following equations on the same coordinate plane and look for a relation between them.
Equations
2x+3y=6
2x+3y=12
2x+3y=18
In order to sketch the graph of the given equations we will find the interception points of them. Let's find the interception points, starting with the first equation.
2x+3y=6
Let's find the x-intercept first.
Since we have found the y-intercept we can sketch the graph, but let's first make a table of interception points for the other equations.
Equations
x=0
y-intercept
y=0
x-intercept
2x+3y=6
2( 0)+3y=6
(0,2)
2x+3( 0)=6
(3,0)
2x+3y=12
2( 0)+3y=12
(0,4)
2x+3( 0)=12
(6,0)
2x+3y=18
2( 0)+3y=18
(0,6)
2x+3( 0)=18
(9,0)
Now we are ready to go! Let's sketch the graphs!
b When we look at the graph in Part A, we can see that their interception points are different. Let's check whether their slopes are the same. To do so, we need to write the equations in slope-intercept form. We will start with the first equation.
Therefore, the slope of the first equation is - 23. We will apply the same process to the other equation and make a table.
Equations
Slope-intercept form
Slope
2x+3y=6
y= - 2/3x+2
- 2/3
2x+3y=12
y= - 2/3x+4
- 2/3
2x+3y=18
y= - 2/3x+6
- 2/3
We can see that the slopes of the lines are equal, so the lines are parallel.
c We will consider what will happen when we increase C in the equation.
2x+3y=C
To make the interpretation easier we will transform the equation into slope-intercept form. From Part B we can write the slope-intercept form of the equation.
y=-2/3x+C/3
From here, we can say that changing the value of C will affect the y-intercept. When we increase C, its graph will shifts upwards through the x- and y-axis. We can observe the changes with the help of the graph as well. When we look at the graph in Part A, we can observe that increasing the values of C shifts the graph upward.
