Match the system of equations to the number of solutions it has.
You can use your Graphing Calculator!
(4 points)
y = 5x + 7
3у - 15x = 18
y=3x+6
y = -13x - 4
x - 2y = 6
3x - 6y = 18
y = 23x - 1
y = 23x - 2
One Solution
No Solution
Infinite Solutions
One Solution:
y = 5x + 7
3у - 15x = 18
No Solution:
y=3x+6
y = -13x - 4
Infinite Solutions:
x - 2y = 6
3x - 6y = 18
One Solution:
y = 23x - 1
y = 23x - 2
To determine the number of solutions for each system of equations, we can use a graphing calculator. Here are the solutions:
1) y = 5x + 7
3у - 15x = 18
This is a system of two linear equations. Graphing them will show that they intersect at a single point. Therefore, the number of solutions is One Solution.
2) y=3x+6
y = -13x - 4
Again, this is a system of two linear equations. Graphing them will show that the lines are parallel and do not intersect. Thus, the number of solutions is No Solutions.
3) x - 2y = 6
3x - 6y = 18
Once more, this is a system of linear equations. However, when graphed, the lines are in the same position and overlap. This means they have an infinite number of points in common, resulting in Infinite Solutions.
4) y = 23x - 1
y = 23x - 2
This system is also composed of linear equations. When graphed, the lines are identical and coincide with each other. Therefore, they have an infinite number of points in common, resulting in Infinite Solutions as well.
To determine the number of solutions for each system of equations, you can use a graphing calculator. Here's how you can do it:
1. Enter the first equation into the graphing calculator: y = 5x + 7.
2. Enter the second equation: 3y - 15x = 18. To do this, you will need to isolate y to get it in the form y = mx + b.
Start with 3y - 15x = 18.
Add 15x to both sides: 3y = 15x + 18.
Divide both sides by 3: y = 5x + 6.
3. Graph the equations on the same graph using the graphing calculator.
a. Press the "y=" button on the calculator.
b. Enter the first equation, y = 5x + 7, into the first line.
c. Enter the second equation, y = 5x + 6, into the second line.
d. Press the "Graph" button to display the graph.
4. Analyze the graph:
- If the lines intersect at one point, the system has one solution. The lines cross at a single point.
- If the lines are parallel and do not intersect, the system has no solution. The lines are equal in slope but have different y-intercepts.
- If the lines overlap, the system has infinite solutions. The equations are equivalent; all points on one line are also on the other.
Now, let's match the given systems of equations to the number of solutions:
1. y = 5x + 7
3y - 15x = 18
Steps:
- Graph the equations using the graphing calculator.
- If the lines intersect at one point, it has one solution.
- If the lines are parallel and do not intersect, it has no solution.
- If the lines overlap, it has infinite solutions.
2. y = 3x + 6
y = -13x - 4
Steps:
- Graph the equations using the graphing calculator.
- If the lines intersect at one point, it has one solution.
- If the lines are parallel and do not intersect, it has no solution.
- If the lines overlap, it has infinite solutions.
3. x - 2y = 6
3x - 6y = 18
Steps:
- Graph the equations using the graphing calculator.
- If the lines intersect at one point, it has one solution.
- If the lines are parallel and do not intersect, it has no solution.
- If the lines overlap, it has infinite solutions.
4. y = 23x - 1
y = 23x - 2
Steps:
- Graph the equations using the graphing calculator.
- If the lines intersect at one point, it has one solution.
- If the lines are parallel and do not intersect, it has no solution.
- If the lines overlap, it has infinite solutions.
By following these steps, you will be able to determine the number of solutions for each system of equations using your graphing calculator.