The solution to a system of two linear equations is x = 3 weeks; y = 9 feet How could you get exactly two or three solutions to this linear system?
To get exactly two or three solutions to a linear system, we need to create a scenario in which the two linear equations intersect at two or three distinct points.
One way to achieve this is by introducing a third equation that shares the same solution as the original system of equations at one or two points. We can do this by adding a parallel line to either one of the original equations.
For example, suppose the original system of equations is:
Equation 1: x + 2y = 15
Equation 2: 3x - 2y = 3
By rearranging Equation 1, we can write it as:
Equation 1 (rearranged): x = 15 - 2y
Now, let's introduce a third equation that shares the same y-intercept as the original system:
Equation 3: x = 15
By adding this equation to the original system, we now have three equations:
Equation 1 (rearranged): x = 15 - 2y
Equation 2: 3x - 2y = 3
Equation 3: x = 15
Now let's solve this system:
1. Substitute Equation 1 (rearranged) into Equation 2:
3(15 - 2y) - 2y = 3
45 - 6y - 2y = 3
45 - 8y = 3
-8y = 3 - 45
-8y = -42
y = -42/-8
y = 5.25
2. Substitute the value of y back into Equation 1 (rearranged):
x = 15 - 2(5.25)
x = 15 - 10.5
x = 4.5
Thus, we have found one solution to this system of linear equations, which is x = 4.5 and y = 5.25.
Now, let's substitute Equation 3 into the original system of equations:
Equation 1 (rearranged): 15 = 15 - 2y
0 = -2y
y = 0
Substituting y = 0 into Equation 1 (rearranged), we get:
x = 15 - 2(0)
x = 15
Therefore, another solution to this system of equations is x = 15 and y = 0.
Hence, by introducing the third equation, we obtained two distinct solutions to the linear system.
Note: If a third equation shares the same solution with the original system at two points instead of one, we would end up with three solutions.
To obtain exactly two solutions to the given linear system, you can use the method of substitution or elimination. Here's a step-by-step explanation of how you can achieve that.
Method 1: Substitution
Step 1: Start with the system of linear equations:
Equation 1: x = 3 weeks
Equation 2: y = 9 feet
Step 2: Pick any arbitrary value for either x or y in one of the equations. Let's say we choose x = 5.
Step 3: Substitute the chosen value of x into the other equation and solve for the other variable. Using equation 1, substitute x = 5:
5 = 3 weeks
Solve for weeks:
weeks = 5 - 3 = 2 weeks
Step 4: Write down the solution as (x, y). In this case, the solution is (x, y) = (5, 9).
Step 5: Repeat the process by picking a different value for x or y and solving for the other variable. For example, you could choose x = 1 and solve for y.
Method 2: Elimination
Step 1: Start with the system of linear equations:
Equation 1: x = 3 weeks
Equation 2: y = 9 feet
Step 2: Multiply one (or both) of the equations by a constant to create an equivalent system that will result in different solutions. Let's multiply equation 1 by 2:
Equation 1: 2x = 6 weeks
Equation 2: y = 9 feet
Step 3: Now, if you subtract the two equations, you will eliminate one variable:
(2x - x) = (6 weeks - 3 weeks)
x = 3 weeks
Step 4: Substitute the value of x back into one of the original equations and solve for the other variable:
3 = 3 weeks
Solve for weeks:
weeks = 3 - 3 = 0 weeks
Step 5: Write down the solution as (x, y). In this case, the solution is (x, y) = (3, 0).
Step 6: Repeat the process by multiplying the equations by different constants, such as -1, to obtain different solutions.
To obtain exactly two solutions or three solutions for a linear system with two equations, you need to modify the equations in a specific way. Let's assume the given system of equations is:
Equation 1: a1x + b1y = c1
Equation 2: a2x + b2y = c2
To get exactly two solutions, you need to make both equations parallel, with identical slopes but different y-intercepts. This means that the coefficients of x and y in both equations need to be proportional. The equations may look like this:
Equation 1: a1x + b1y = c1
Equation 2: ka1x + kb1y = kc1
Here, k is a non-zero constant. By multiplying Equation 1 by 'k,' you create an equivalent equation with the same slope but a different y-intercept.
For example, let's consider the given solution: x = 3 weeks and y = 9 feet. We can express it as:
Equation 1: x - 3 = 0
Equation 2: x - 3 = 0
This system of equations has an infinite number of solutions because the equations are identical.
To get exactly three solutions, you need to make the two equations intersect at two different points, resulting in three distinct solutions. To achieve this, the equations need to have different slopes and different y-intercepts. The equations may look like this:
Equation 1: a1x + b1y = c1
Equation 2: a2x + b2y = c2
Here, the coefficients of x and y in both equations should not be proportional. This will ensure that the two lines intersect at two different points.
For example, consider the following equations:
Equation 1: x - 3 = 0
Equation 2: y - 9 = 0
These equations represent two lines that intersect at the point (3, 9), which gives a unique solution.
In summary, to obtain exactly two solutions for a linear system, make the equations parallel by multiplying one equation by a non-zero constant. To obtain exactly three solutions, make the equations intersect at two different points by ensuring the slopes and y-intercepts are different.