Answer:
- Pair 1 has one solution.
- Pair 2 has one solution.
- Pair 3 has infinite solutions.
- Pair 4 has no solution.
- Pair 5 has one solution.
Explanation:
To determine which pairs of equations form a system with one solution, no solution, or infinite solutions, we need to solve each equation and check their slopes and y-intercepts.
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. 2y = 6x + 2
5. y - 5x = 2
6. y - 4x + 1 = 0
Now, let's analyze each equation:
1. Equation 1: y = 5x - 4
This equation is already in slope-intercept form (y = mx + b), where m represents the slope (5) and b represents the y-intercept (-4).
2. Equation 2: y = -2x + 3
This equation is also in slope-intercept form, where m represents the slope (-2) and b represents the y-intercept (3).
3. Equation 3: y = 3x + 1
This equation is also in slope-intercept form, where m represents the slope (3) and b represents the y-intercept (1).
4. Equation 4: 2y = 6x + 2
To put this equation in slope-intercept form, we need to isolate y:
Divide both sides of the equation by 2:
y = 3x + 1
This equation is equivalent to Equation 3, so it represents the same line.
5. Equation 5: y - 5x = 2
To put this equation in slope-intercept form, we need to isolate y:
Add 5x to both sides of the equation:
y = 5x + 2
This equation is not the same as any of the previous equations.
6. Equation 6: y - 4x + 1 = 0
To put this equation in slope-intercept form, we need to isolate y:
Add 4x and 1 to both sides of the equation:
y = 4x - 1
This equation is not the same as any of the previous equations.
Now, let's analyze the pairs of equations:
Pair 1: Equations 1 and 2
These equations represent different lines with different slopes (-2 and 5). Therefore, they intersect at one point, forming a system with one solution.
Pair 2: Equations 1 and 3
These equations represent different lines with different slopes (3 and 5). Therefore, they intersect at one point, forming a system with one solution.
Pair 3: Equations 1 and 4
Equation 4 is equivalent to Equation 3. Therefore, these equations represent the same line. They intersect at every point on the line, forming a system with infinite solutions.
Pair 4: Equations 1 and 5
These equations represent different lines with different slopes (5 and 5). However, their y-intercepts are different (-4 and 2). Therefore, they do not intersect, forming a system with no solution.
Pair 5: Equations 1 and 6
These equations represent different lines with different slopes (4 and 5). Therefore, they intersect at one point, forming a system with one solution.
I hope this helps!