Find the number of the solutions to each system.

a. 4x-y+1=0 4x-y+3=0
b. 2x-y+4=0 4x-2y+8=0
Write a question that can be solved using a system of linear equations.
Solve the following system of equations by graphing: y=x+3 and y=-2x-3

(a) no solutions. How can 4x-y be both -1 and -3?

(b) many solutions: if you multiply the first equation by 2, you will see that both equations describe the same line.

Google is your friend. Use it to get lots of examples of linear equations in problems.

To solve by graphing, you'll have to do the graphs. The solution is where the graphed lines intersect.

In google type :

functions graphs online

When you see list of results click on :

Draw Function Graphs - Plotter - Rechneronline

When page be open in blue rectangle type :

x + 3

In gray rectangle type :

- 2 x - 3

Then click option :

Draw

Solution :

x = - 2 , y = 1

a. To find the number of solutions to the system of equations 4x-y+1=0 and 4x-y+3=0, we need to determine if the two equations represent parallel lines or if they intersect at a single point.

To do this, we can rewrite the given equations in slope-intercept form (y = mx + b):

Equation 1: 4x - y + 1 = 0 -> y = 4x + 1
Equation 2: 4x - y + 3 = 0 -> y = 4x + 3

Comparing the slopes of both equations, we can see that the slopes are identical (both are equal to 4). Therefore, the lines are parallel and will never intersect. Thus, the system has no solution.

b. To find the number of solutions to the system of equations 2x-y+4=0 and 4x-2y+8=0, we follow a similar approach as in part a.

Equation 1: 2x - y + 4 = 0 -> y = 2x + 4
Equation 2: 4x - 2y + 8 = 0 -> 2y = 4x + 8 -> y = 2x + 4

Since both equations have the same slope (2), the lines are parallel and will never intersect. Thus, the system has no solution.

A question that can be solved using a system of linear equations is:

A bakery sells two types of pastries: muffins and cookies. On Monday, they sold a total of 60 pastries, and the revenue from those sales was $240. Muffins cost $4 each, and cookies cost $3 each. How many muffins and how many cookies were sold on Monday?

To find the number of solutions for a system of linear equations, you need to check the relationship between the coefficients of the variables. Let's analyze each system individually:

a. 4x - y + 1 = 0 and 4x - y + 3 = 0

In this system, you can see that both equations have the same coefficients for x and y, which means they represent the same line. Therefore, the two equations are essentially the same and would coincide when plotted on a graph. So, they have infinitely many solutions.

b. 2x - y + 4 = 0 and 4x - 2y + 8 = 0

Here, if we compare the coefficients, we find that the ratio between the coefficients of x in the two equations is not equal to the ratio between the coefficients of y. Thus, the lines represented by these equations are not parallel or coincident. Therefore, this system has a unique solution.

Now let's create a question that can be solved using a system of linear equations:

Question: Bob has $20 in coins consisting of quarters (25 cents) and dimes (10 cents). If he has a total of 100 coins, how many quarters and how many dimes does he have?

To solve this question, we can set up a system of linear equations. Let's assume 'x' represents the number of quarters and 'y' represents the number of dimes. The two equations will be:

1. 0.25x + 0.10y = 20 (equation representing the total value of coins)
2. x + y = 100 (equation representing the total number of coins)

By solving this system of equations, we can determine the number of quarters and dimes Bob has.