Is it possible to find a value for c so that the linear system below has exactly one solution? Explain.
5x+3y= 21 Equation 1
y= -5/3x+c Equation 2
1) 5x+3y = 21
2) 5x+3y = 3c
since the two lines have the same slope, they either coincide or never intersect.
No unique solution exists.
To find a value of c that will give the linear system exactly one solution, we need to determine if the two equations intersect at a single point.
First, let's rewrite equation 2 in terms of y:
y = -5/3x + c
Now, we can substitute this expression for y into equation 1:
5x + 3(-5/3x + c) = 21
Simplifying the equation:
5x - 5x + 3c = 21
3c = 21
c = 7
Now, let's substitute c = 7 back into equation 2:
y = -5/3x + 7
So, the value of c that will give the linear system exactly one solution is c = 7. This means that when c = 7, the two lines will intersect at a single point, and there will be a unique solution.
To determine if the linear system has exactly one solution, we need to check for a condition called "consistent and independent."
A linear system is consistent if there is at least one solution, and it is independent if there is exactly one solution. In other words, if the equations in the system describe two lines, they need to intersect at exactly one point.
Now let's analyze the given equations and see if we can find a value for c that satisfies the condition.
Equation 1: 5x + 3y = 21
Equation 2: y = (-5/3)x + c
To find the intersection point between the two lines, we can equate the right-hand sides of both equations and solve for x and y.
(-5/3)x + c = y (equating the right-hand sides of both equations)
Substituting this value of y into Equation 1:
5x + 3((-5/3)x + c) = 21
5x - 5x + 3c = 21
3c = 21
From this equation, we can determine the value of c by dividing both sides by 3:
c = 21/3
c = 7
Therefore, if c is equal to 7, the linear system will have exactly one solution.