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.
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.
11 answers
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.
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.
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
- 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.
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.
The x and y coordinates
To find the x and y coordinates for the different pairs of equations, we need to solve for both x and y in each equation. Let's list them out:
1. Equations 1 and 2:
- Equation 1: y = 5x - 4
x-coordinate: 1, y-coordinate: 1 (Solution: (1, 1))
2. Equations 3 and 4:
- Equation 3: y = 3x + 1
Infinite solutions
3. Equations 5 and 6:
- Equation 5: y = 5x + 2
x-coordinate: -3, y-coordinate: -13 (Solution: (-3, -13))
4. Equations 1 and 7:
- Equation 1: y = 5x - 4
No solution
These are the x and y coordinates corresponding to the solution for each pair of equations as discussed earlier.
1. Equations 1 and 2:
- Equation 1: y = 5x - 4
x-coordinate: 1, y-coordinate: 1 (Solution: (1, 1))
2. Equations 3 and 4:
- Equation 3: y = 3x + 1
Infinite solutions
3. Equations 5 and 6:
- Equation 5: y = 5x + 2
x-coordinate: -3, y-coordinate: -13 (Solution: (-3, -13))
4. Equations 1 and 7:
- Equation 1: y = 5x - 4
No solution
These are the x and y coordinates corresponding to the solution for each pair of equations as discussed earlier.
3 answers