How many solutions are there for the following
system of equations?
y= X - 5x+3
y= X- 6
To find the number of solutions, we can set the two equations equal to each other because they both equal y:
X - 5x + 3 = X - 6
Subtracting X from both sides gives us:
-5x + 3 = -6
Subtracting 3 from both sides gives us:
-5x = -9
Dividing by -5 gives us:
x = 9/5
Now we can plug this value of x into either of the original equations to find the corresponding y value. Using the second equation, we get:
y = (9/5) - 6 = -21/5
Therefore, there is only one solution to this system of equations: (9/5, -21/5).
To determine the number of solutions for this system of equations, we can analyze the relationship between the two equations.
The given equations are:
1) y = X - 5x + 3
2) y = X - 6
To find the number of solutions, we need to examine the slopes and y-intercepts of the equations.
Let's rearrange both equations into the slope-intercept form (y = mx + b), where m represents the slope and b represents the y-intercept:
1) y = -5x + x + 3
Simplifying, we get: y = -4x + 3
2) y = x - 6
From these equations, we can determine that the slope of the first equation is -4, while the slope of the second equation is 1.
If the slopes are different, as they are in this case, then the lines represented by the equations will intersect at a single point, resulting in a unique solution. This means that there is exactly one solution for this system of equations.
To find the solutions for this system of equations, we need to equate the expressions for y and solve for x.
Given:
y = x - 5x + 3 (equation 1)
y = x - 6 (equation 2)
To simplify equation 1, we combine like terms:
y = -4x + 3
Now we have the following system of equations:
y = -4x + 3 (equation 1)
y = x - 6 (equation 2)
Since both equations are set equal to y, we can set them equal to each other:
-4x + 3 = x - 6
To solve for x, we can bring all the terms with x to one side and the constants to the other side:
-4x - x = -6 - 3
-5x = -9
Dividing both sides by -5:
x = -9 / -5
x = 9/5
Now, substitute the value of x back into either equation to find y.
Let's use equation 2:
y = (9/5) - 6
y = 9/5 - 30/5
y = -21/5
Therefore, the solution to this system of equations is x = 9/5 and y = -21/5. There is only one solution.