Solved on Sep 25, 2023

Graph the linear equations in slope-intercept form: y=3x+5y=3x+5 and y=3x+83y=3x+\frac{-8}{3}. Determine if the system is consistent or inconsistent, and if consistent, provide the solution. Determine if the equations are dependent or independent.

STEP 1

Assumptions1. We are given two linear equations y=3x+5y =3x +5 y=3x83y =3x - \frac{8}{3}. We need to graph these equations and determine whether the system is consistent or inconsistent.
3. We also need to determine whether the equations are dependent or independent.

STEP 2

First, let's observe the given equations. They are already in slope-intercept form, which is y=mx+by = mx + b, where mm is the slope and bb is the y-intercept.
The first equation y=x+5y =x +5 has a slope of $$ and a y-intercept of $5$.
The second equation y=x8y =x - \frac{8}{} also has a slope of $$ and a y-intercept of $-\frac{8}{}$.

STEP 3

To graph these equations, we start by plotting the y-intercepts on the y-axis. Then, we use the slope to find another point for each line. The slope, being the ratio of the vertical change to the horizontal change between any two points on the line, tells us to move3 units up for every1 unit we move to the right.

STEP 4

After plotting the lines, we observe that they are parallel to each other. This is because they have the same slope but different y-intercepts.

STEP 5

A system of linear equations is consistent if it has at least one solution, and inconsistent if it has no solutions. Since these two lines are parallel and do not intersect, there are no points that satisfy both equations. Therefore, the system is inconsistent.

STEP 6

A system of equations is dependent if all solutions of one equation are also solutions of the other equation. In other words, the equations represent the same line. Independent equations, on the other hand, represent different lines. Since our two equations represent different lines, they are independent.
The solution to the problem is that the system of equations is inconsistent and the equations are independent.

Was this helpful?
banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ContactInfluencer programPolicyTerms
TwitterInstagramFacebookTikTokDiscord