Examine the different equations that follow:
1. y=5x–4
2. y=–2x+3
3. y=3x+1
4. 2y=6x+2
5. y–5x=+2
6. y–4x+1=0
7. y=5x
Complete steps a. through e. to determine which pair of equations form a system
with one solution, no solution, or infinite solutions.
a. Solve each equation and make sure it is in y=mx+b format.
1. y=5x-4
2. y=-2x+3
3. y=3x+1
4. y=3x+1 (after dividing by 2)
5. y=5x+2 (after rearranging terms)
6. y=4x-1 (after rearranging terms)
7. y=5x
b. Determine the slope (m) and y-intercept (b) for each equation:
1. Slope = 5, y-intercept = -4
2. Slope = -2, y-intercept = 3
3. Slope = 3, y-intercept = 1
4. Slope = 3, y-intercept = 1
5. Slope = 5, y-intercept = 2
6. Slope = 4, y-intercept = -1
7. Slope = 5, y-intercept = 0
c. Plot each line on a graph to visually see if they intersect at one point, are parallel, or overlap (infinite solutions).
d. Compare the slopes of each pair of equations:
1. Slope = 5
2. Slope = -2
3. Slope = 3
4. Slope = 3
5. Slope = 5
e. Compare the y-intercepts of each pair of equations and determine if the lines intersect at that point:
1. y-intercept = -4
2. y-intercept = 3
3. y-intercept = 1
4. y-intercept = 1
5. y-intercept = 2
By comparing the slopes and y-intercepts of each pair of equations, we can determine if they have one solution, no solution, or infinite solutions.
the answers to the equations are
To determine the number of solutions for each pair of equations:
1. Equations 1 and 2:
- Equation 1: y=5x-4
- Equation 2: y=-2x+3
- These equations have one solution as they have different slopes and intersect at a single point.
2. Equations 3 and 4:
- Equation 3: y=3x+1
- Equation 4: 2y=6x+2 -> y=3x+1
- These equations are actually the same line and have infinite solutions since they are coincident.
3. Equations 5 and 6:
- Equation 5: y=5x+2
- Equation 6: y=4x-1
- These equations have no solution as they have different slopes and do not intersect.
4. Equations 1 and 7:
- Equation 1: y=5x-4
- Equation 7: y=5x
- These equations have no solution as one is a line with a y-intercept and the other is a line passing through the origin.
Therefore, the system with one solution is Equations 1 and 2, the system with infinite solutions is Equations 3 and 4, and the systems with no solutions are Equations 5 and 6 as well as Equations 1 and 7.
the number answers
Based on the provided equations, the number of solutions for each pair is as follows:
1. Equations 1 and 2: One solution
2. Equations 3 and 4: Infinite solutions
3. Equations 5 and 6: No solution
4. Equations 1 and 7: No solution
Thank you for the clarification.
the solutions
1. Equations 1 and 2:
- y = 5x - 4
- y = -2x + 3
To find the solution, we set the equations equal to each other:
5x - 4 = -2x + 3
Rearranging terms gives us:
7x = 7
x = 1
Plugging x back into one of the equations (e.g., y = 5(1) - 4), we find y = 1.
Therefore, the solution to Equations 1 and 2 is (1, 1).
2. Equations 3 and 4:
- y = 3x + 1
- y = 3x + 1
Since the equations are the same, they have an infinite number of solutions.
3. Equations 5 and 6:
- y - 5x = 2
- y = 4x - 1
To find the solution, we substitute the value of y from the first equation into the second equation:
5x + 2 = 4x - 1
Rearranging terms gives us:
x = -3
Plugging x back into the first equation (y = 5(-3) + 2), we find y = -13.
Therefore, the solution to Equations 5 and 6 is (-3, -13).
4. Equations 1 and 7:
- y = 5x - 4
- y = 5x
Since these equations represent parallel lines (same slope, different y-intercepts), they have no solution.
Therefore, the solutions to the pairs of equations are:
1. Equations 1 and 2: (1, 1)
2. Equations 3 and 4: Infinite solutions
3. Equations 5 and 6: (-3, -13)
4. Equations 1 and 7: No solution
coordinates for slope
Sure, here are the slopes of the given equations:
1. Equation 1: y = 5x - 4
Slope = 5
2. Equation 2: y = -2x + 3
Slope = -2
3. Equation 3: y = 3x + 1
Slope = 3
4. Equation 4: y = 3x + 1
Slope = 3
5. Equation 5: y = 5x + 2
Slope = 5
6. Equation 6: y = 4x - 1
Slope = 4
7. Equation 7: y = 5x
Slope = 5
These are the slopes for each of the given equations.