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
This is a linear system of equations.
The first equation y = 5x + 7 represents a line with a slope of 5 and a y-intercept of 7.
The second equation 3y - 15x = 18 can be simplified to y - 5x = 6, which represents another line with the same slope of 5.
Since both equations represent lines with the same slope but different y-intercepts, they will intersect at a single point, indicating that this system of equations has one solution.
To determine the number of solutions for the system of equations, we can look at the slopes of the two lines.
The given equation y = 5x + 7 has a slope of 5.
The equation 3y - 15x = 18 can be rearranged to y = 5x + 6, which also has a slope of 5.
Since both equations have the same slope, the lines are parallel, and the system will have no solutions.
To determine the number of solutions for a system of equations, we can use graphing calculator or analyze the equations.
Let's start by graphing the first equation, y = 5x + 7:
1. Press the "y=" button on your graphing calculator.
2. Enter the equation y = 5x + 7 into the first line.
3. Press the "Graph" button to plot the equation.
Now let's graph the second equation, 3y - 15x = 18:
1. Press the "y=" button again.
2. Solve the equation for y: 3y = 15x + 18 (divide both sides by 3) => y = 5x + 6.
3. Enter the equation y = 5x + 6 into the second line.
4. Press the "Graph" button to plot the equation.
Now, let's analyze the graph on your calculator:
- If the lines intersect at one point, the system of equations has one unique solution. In other words, the lines cross each other at a single point.
- If the lines are parallel and do not intersect, the system has no solutions.
- If the lines overlap exactly, the system has infinitely many solutions.
By looking at the graph on your graphing calculator, you should be able to determine the type of solutions for this system of equations.