Match the system of equations to the number of solutions it has.
You can use your Graphing Calculator!
(4 points)
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y = 5x + 7
3y - 15x = 18
y = 3x + 6
y = -13x - 4
x - 2y = 6
3x - 6y = 18
y = 23x - 1
y = 23x - 2
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y = 5x + 7 and 3y - 15x = 18: One solution
y = 3x + 6 and y = -13x - 4: One solution
x - 2y = 6 and 3x - 6y = 18: Infinitely many solutions
y = 23x - 1 and y = 23x - 2: No solution
To determine the number of solutions for each system of equations, we can use a graphing calculator. Here's how:
1. Enter the first equation, y = 5x + 7, into the graphing calculator. Graph the equation.
2. Enter the second equation, 3y - 15x = 18, into the graphing calculator. Graph the equation.
3. Locate the point of intersection between the two graphs.
- If the graphs intersect at a single point, the system of equations has one solution.
- If the graphs are parallel and do not intersect, the system of equations has no solution.
- If the graphs overlap and coincide with each other, the system of equations has infinitely many solutions.
Repeat the above steps for each set of equations given, and match the number of solutions to the corresponding system of equations.
To determine the number of solutions for each system of equations, you can simplify the equations and compare the coefficients.
System 1:
y = 5x + 7
3y - 15x = 18
System 2:
y = 3x + 6
y = -13x - 4
System 3:
x - 2y = 6
3x - 6y = 18
System 4:
y = 23x - 1
y = 23x - 2
Let's analyze each system one by one:
System 1:
Comparing the coefficients, we can see that the first equation in System 1 is the same as the second equation in System 2. Therefore, these two systems represent the same line and have infinitely many solutions.
System 2:
Comparing the coefficients, we can see that the slopes of the two lines are different (-13 and 3), so the lines will intersect at exactly one point. Therefore, System 2 has a unique solution.
System 3:
Comparing the coefficients, we can see that the slopes of the two lines are the same (1 and 3), but the y-intercepts are different. Therefore, the lines are parallel and do not intersect. System 3 has no solution.
System 4:
Comparing the coefficients, we can see that the equations are identical. Therefore, System 4 represents a single line and has infinitely many solutions.
To summarize:
System 1 has infinitely many solutions.
System 2 has a unique solution.
System 3 has no solution.
System 4 has infinitely many solutions.