What information can the slope-intercept form of an equation give us?
D
Multiply each side of the first equation by - 2 and compare the result with the second equation.
A
The lines intersect because their slopes are different.
B
The lines are parallel because the system has no solution.
C
The lines are parallel because they have the same slope but different y-intercepts.
D
The lines coincide because the equations are equivalent.
Practice makes perfect
Let's consider the given system of equations.
y & = 2x+3 y & = 12x+2
We want to determine by inspection whether the lines in the system intersect, coincide, or are parallel. Let's start by noticing that the equations are written in slope-intercept form. We can use this form to identify the key features of the graphs of the equation.
Slope-Intercept Form
Slope m
y-intercept b
y= 2x+ 3
2
(0, 3)
y= 1/2x+ 2
1/2
(0, 2)
Since the slopes of the lines are different, the lines cannot be parallel and do notcoincide. This means that the lines intersect.
We want to determine by inspection whether the lines represented by the given system of equations intersect, coincide, or are parallel. Let's start by analyzing the equations!
2x+3y & = 6 2x+3y & = 9
According to the first equation, the expression 2x+3y is equal to 6. In the second equation, the same expression is equal to 9. This means that the system of equation has no solutions and represents two different lines that do notintersect. Therefore, the lines are parallel.
Let's take a look at the given system of equations.
y & = 13x+2 y & = 13x-2
We want to figure out whether the lines in the system intersect, coincide, or are parallel. Let's start by using the slope-intercept form of each equation to identify the key features of their graphs.
Equation
Slope-Intercept Form
Slope m
y-intercept b
y=1/3x+2
y= 1/3x+ 2
1/3
(0, 2)
y=1/3x-2
y= 1/3x+( - 2)
1/3
(0, - 2)
We can see that the y-intercepts of the lines are different, which means that the lines do notcoincide. Since the lines have the same slope and do notcoincide, they are parallel.
We want to figure out whether the lines represented by the given system of equations intersect, coincide, or are parallel.
x-2y & = 4 - 2x+4y & = - 8
Notice that if we multiply each side of the first equation by - 2, we will be able to easily compare the equations. Let's do it!
We found that the first equation is equivalent to the second one. This means that the equations represent the same line. In other words, the lines coincide.
