How many solutions does the system of equations have. -5x+y=-18. 10x-2y=36
We can solve this system of equations by using the method of substitution or the method of elimination.
Let's solve it using the method of elimination:
Multiply the first equation by 2 and the second equation by -1 to cancel out the y terms:
-10x + 2y = -36
+ 10x - 2y = -36
When we add these equations together, we get:
0 = -72
Since the equation is false (-72 is not equal to 0), there is no solution for this system of equations.
To solve this system of equations, we can use the method of elimination. We will multiply the first equation by 2 and the second equation by 1 to get rid of the y term:
Equation 1: 2*(-5x+y) = 2*(-18) => -10x+2y = -36
Equation 2: 1*(10x-2y) = 1*(36) => 10x-2y = 36
Now we can add the two equations to eliminate the y term:
(-10x+2y) + (10x-2y) = -36 + 36
-10x+2y + 10x-2y = 0
0 = 0
The result is 0 = 0, which means there is an infinite number of solutions. The two equations represent the same line, so every point on the line is a solution to the system.
To determine how many solutions the system of equations has, we can solve the equations simultaneously and analyze the result. Let's go through the process step by step.
Given System of Equations:
-5x + y = -18 [Equation 1]
10x - 2y = 36 [Equation 2]
Step 1: Solve Equation 1 for y.
Rearrange Equation 1 to isolate y:
y = 5x - 18 [Equation 1a]
Step 2: Substitute Equation 1a into Equation 2.
Replace y in Equation 2 with its equivalent expression from Equation 1a:
10x - 2(5x - 18) = 36
10x - 10x + 36 = 36
36 = 36
Step 3: Analyze the result.
In Step 2, we derived the equation "36 = 36." This equation states that 36 is equal to 36, which means the equation is always true regardless of the value of x.
Since the equation is always true, it indicates that both original equations are equivalent, resulting in infinitely many solutions. In other words, every point on the line represented by the equations satisfies the system.
Therefore, the system of equations has infinitely many solutions.