Match the system of equations to the number of solutions it has. You can use your Graphing Calculator! (4 points) Put responses in the correct input to answer the question. Select a response%2C navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar%2C enter key%2C left mouse button or touchpad. Responses can also be moved by dragging with a mouse. y %3D 5x %2B 7 3y - 15x %3D 18 y %3D 3x %2B 6 y %3D -13x - 4 x - 2y %3D 6 3x - 6y %3D 18 y %3D 23x - 1 y %3D 23x - 2
y = 5x + 7: Infinite solutions
3y - 15x = 18: One solution
y = 3x + 6: One solution
y = -13x - 4: One solution
x - 2y = 6: One solution
3x - 6y = 18: One solution
y = 23x - 1: One solution
y = 23x - 2: One solution
To determine the number of solutions each system of equations has, we can use a graphing calculator. Here's how:
1. Enter the first equation into the graphing calculator: y = 5x + 7
2. Graph the equation. This will give you a straight line on the graph.
3. Repeat the above steps for the remaining equations: y = 3x + 6, y = -13x - 4, x - 2y = 6, 3x - 6y = 18, y = 23x - 1, y = 23x - 2
4. Once you have graphed all the equations, observe the intersections on the graph.
Based on the intersections of the equations, match each system of equations to the number of solutions it has:
1. y = 5x + 7
2. 3y - 15x = 18
This system has __1__ solution.
3. y = 3x + 6
4. x - 2y = 6
This system has __1__ solution.
5. 3x - 6y = 18
6. y = -13x - 4
This system has __1__ solution.
7. y = 23x - 1
8. y = 23x - 2
This system has __infinitely many__ solutions.
Now you can match the systems of equations to the number of solutions they have using the responses provided.
To determine the number of solutions for each system of equations, we can use the following rules:
1. If the lines represented by the equations have different slopes, the system has exactly one solution.
2. If the lines are parallel (have the same slope) and different y-intercepts, the system has no solution.
3. If the lines are identical, or coincident, the system has infinitely many solutions.
Let's analyze each system of equations:
1. y = 5x + 7
3y - 15x = 18
2. y = 3x + 6
y = -13x - 4
3. x - 2y = 6
3x - 6y = 18
4. y = 23x - 1
y = 23x - 2
By observing the equations, we can determine the following:
1. The slopes of the lines in the first system are different, so it has exactly one solution.
2. The slopes of the lines in the second system are different, so it also has exactly one solution.
3. The slopes of the lines in the third system are the same, but the y-intercepts are different. Therefore, it has no solution.
4. The slopes of the lines in the fourth system are the same, and the y-intercepts are the same too. This means the lines are identical, so it has infinitely many solutions.
To summarize:
1. System 1 has exactly one solution.
2. System 2 has exactly one solution.
3. System 3 has no solution.
4. System 4 has infinitely many solutions.
Judy has %24100 and Elenore has %2475. Judy is saving %245 per day and Elenore is saving %247 per day. (4 points) What equation represent how much money Judy will have in total after saving for d days%3F What equation represents how much money Elenore will have in total after saving for d days%3F What equation would be used to find out how many days it will take for Judy and Elenore to have the same amount of money saved%3F After the number of days it took both to have the same amount of money saved%2C how much money do they have each%3F
The equation that represents how much money Judy will have in total after saving for d days is:
Judy's total savings = $100 + ($5 × d)
The equation that represents how much money Elenore will have in total after saving for d days is:
Elenore's total savings = $75 + ($7 × d)
The equation that can be used to find out how many days it will take for Judy and Elenore to have the same amount of money saved is:
Judy's total savings = Elenore's total savings
$100 + ($5 × d) = $75 + ($7 × d)
To solve this equation and find the number of days it will take for Judy and Elenore to have the same amount of money saved, we can subtract $75 from both sides and then divide by the difference in saving rates:
$100 - $75 = ($7 × d) - ($5 × d)
$25 = $2 × d
25/2 = d
After the number of days it took both to have the same amount of money saved (d = 12.5 days), we can substitute this value into either of the original equations to find out how much money they each have:
Judy's total savings = $100 + ($5 × 12.5) = $100 + $62.5 = $162.5
Elenore's total savings = $75 + ($7 × 12.5) = $75 + $87.5 = $162.5
So, after 12.5 days, both Judy and Elenore have $162.5 each.