Consider the following system of equations.
y = −2x − 3
y = −2x + 2
Are the graphs of the two lines intersecting lines, the same line, or parallel lines? Explain your reasoning.
How many solutions does the system have? Explain how you can tell without solving the system.
Solve the system using any method. Show your work.
3x − 2y = 10
−1.5x + y = 5
in y = mx + b , (both of your equations are in that form)
m stands for the slope, and b is the y-intercept
They both have the same slope, so they must be parallel
One cuts the y-axis at -3, the other at 2
So what do you think???
for you second set of equations, arrange them in the form y = mx + b
to make your decision
The system of linear equations:
y = −2x − 3
y = −2x + 2.
What type of line (parallel, intersecting, and same) does this equation express?
Parallel Line: Same slope; Different y-intercepts.
Intersecting Line: Different slope; Different y-intercepts.
Same Line: Same slope; Same y-intercepts.
It is a parallel line since the slopes are the same, but the y-intercepts are different. Furthermore, it has no solution because parallel lines have a y-intercept (which is -1), but never has an intersection. Y=mx+b is a slope-intercept formula that may be used to solve the problem by graphing the lines.
To determine whether the graphs of the two lines are intersecting, the same line, or parallel lines, we can compare their slopes.
Let's represent the equations in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.
For the first equation, y = -2x - 3, the slope is -2.
For the second equation, y = -2x + 2, the slope is -2.
Since both equations have the same slope, -2, the lines are parallel.
Since parallel lines do not intersect, the system of equations has no solutions.
To solve the system using any method, let's use the method of substitution:
1. Solve the second equation for y in terms of x:
-1.5x + y = 5
y = 1.5x + 5
2. Substitute this value of y into the first equation:
3x - 2(1.5x + 5) = 10
3x - 3x - 10 = 10
-10 = 10
This equation is always false. Therefore, there are no values of x and y that satisfy both equations, and the system has no solution.
To determine whether the system of equations has intersecting lines, the same line, or parallel lines, we need to compare the slopes and y-intercepts of both equations.
The given system of equations:
y = -2x - 3 equation (1)
y = -2x + 2 equation (2)
Both equations have the same slope (-2) but different y-intercepts (-3 and +2).
If two lines have the same slope and different y-intercepts, they are parallel lines and do not intersect.
Therefore, the graphs of the two lines in this system are parallel lines.
Now, let's determine the number of solutions in the system without solving it.
When two lines are parallel, they do not intersect, so the system has no solution. In other words, there are no values of x and y that satisfy both equations simultaneously.
Therefore, the given system has no solution.
To solve the system using any method, let's use the method of substitution:
Given system:
3x - 2y = 10 equation (1)
-1.5x + y = 5 equation (2)
Step 1: Solve equation (2) for y:
-1.5x + y = 5
y = 1.5x + 5 equation (3)
Step 2: Substitute equation (3) into equation (1):
3x - 2(1.5x + 5) = 10
3x - 3x - 10 = 10
-10 = 10
Since -10 does not equal 10, we have obtained an equation that is not true. This means that the system of equations is inconsistent, indicating that the system has no solution.
Therefore, the given system has no solution.