Question 1
Examine the six 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
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 (already in y=mx+b format)
2. y = -2x + 3 (already in y=mx+b format)
3. y = 3x + 1 (already in y=mx+b format)
4. 2y = 6x + 2
Divide both sides by 2 to isolate y:
y = 3x + 1 (now in y=mx+b format)
5. y - 5x = 2
Add 5x to both sides to isolate y:
y = 5x + 2 (now in y=mx+b format)
6. y - 4x + 1 = 0
Subtract 4x and 1 from both sides to isolate y:
y = 4x - 1 (now in y=mx+b format)
Now all the equations are in y=mx+b format.
what ones have no solution, one solution, or infinite solutions
To determine which pair of equations have no solution, one solution, or infinite solutions, we need to compare their slopes and y-intercepts.
1. Equation: y = 5x - 4
Slope (m): 5
Y-intercept (b): -4
2. Equation: y = -2x + 3
Slope (m): -2
Y-intercept (b): 3
3. Equation: y = 3x + 1
Slope (m): 3
Y-intercept (b): 1
4. Equation: y = 3x + 1
Slope (m): 3
Y-intercept (b): 1
5. Equation: y = 5x + 2
Slope (m): 5
Y-intercept (b): 2
6. Equation: y = 4x - 1
Slope (m): 4
Y-intercept (b): -1
Now let's compare the equations:
- Equations 1 and 4 have the same slope and same y-intercept. They represent the same line, so they have infinite solutions.
- Equations 2, 3, 5, and 6 have different slopes and different y-intercepts. They represent different lines, so each pair of equations has one solution.
you need to solve those equations
Apologies for misunderstanding your request. Let's solve each equation to determine the solutions.
1. y = 5x - 4
To find the solution, we need another equation to form a system. Let's move on to the next equation.
2. y = -2x + 3
Again, we need another equation to form a system.
3. y = 3x + 1
Similar to the previous equations, we need another equation to solve the system.
4. 2y = 6x + 2
Divide both sides by 2:
y = 3x + 1
Now we have two equations to form the system:
y = 5x - 4
y = 3x + 1
To find the solution, we can set them equal to each other:
5x - 4 = 3x + 1
Subtract 3x from both sides:
2x - 4 = 1
Add 4 to both sides:
2x = 5
Divide both sides by 2:
x = 2.5
Now, substitute x back into either equation to find y:
y = 3(2.5) + 1
y = 7.5 + 1
y = 8.5
Therefore, the solution to the system of equations is x = 2.5 and y = 8.5. This system has one solution.
Let's continue with the other equations to find their solutions.
5. y - 5x = 2
Solve for y in terms of x:
y = 5x + 2
Again, we need another equation to form a system.
6. y - 4x + 1 = 0
Rearrange the equation:
y = 4x - 1
Similarly, we need another equation to form a system.
Based on the given equations, only equation 4 (2y = 6x + 2) and equation 3 (y = 3x + 1) form a system with one solution. The other equations either require additional equations to form a system or have no solution.
To determine the solutions for each equation and convert them into y=mx+b format, follow these steps:
Step a: Solve each equation for y
1. y = 5x - 4
2. y = -2x + 3
3. y = 3x + 1
4. 2y = 6x + 2 (To isolate y, divide both sides by 2)
y = 3x + 1
5. y - 5x = 2 (Add 5x to both sides)
y = 5x + 2
6. y - 4x + 1 = 0 (Move 4x and 1 to the other side)
y = 4x - 1
Now all the equations are in the y=mx+b format.
Equations in y=mx+b format:
1. y = 5x - 4
2. y = -2x + 3
3. y = 3x + 1
4. y = 3x + 1
5. y = 5x + 2
6. y = 4x - 1
Step b: Now that the equations are in the same format, we can compare them to determine the type of system they form.
Step c: To identify a system with one solution, we need to look for a pair of equations with different slopes (m-values). If the slopes are different, the lines intersect at a single point.
Step d: To identify a system with no solution, we need to look for a pair of equations with the same slopes but different y-intercepts (b-values). If the lines have the same slope but different y-intercepts, they are parallel and do not intersect.
Step e: To identify a system with infinite solutions, we need to look for a pair of equations that are equivalent. Equations that are equivalent represent the same line and have infinitely many points in common.
Now you can compare the equations to determine the type of system.