describe the solution to this system of equations:
8x-2y=-4
4x-y=-2
A. solve this system by elimination since it is already in standard form and lined up nicely.
B. solve this system by subsitution since one of the variables is defined by the other without having to do any math.
The solution to this system of equations can be found by using the method of substitution.
First, we can solve one of the equations for one variable in terms of the other. Let's solve the second equation for y:
4x - y = -2
y = 4x + 2
Now, we substitute this expression for y into the first equation:
8x - 2(4x + 2) = -4
8x - 8x - 4 = -4
-4 = -4
Both sides of the equation are equal, which means that the system of equations is consistent and the two equations represent the same line. Therefore, there are infinitely many solutions to this system, and the solution is all values of x and y that satisfy the equations.
In conclusion, the solution to the system of equations is B. solve this system by substitution since one of the variables is defined by the other without having to do any math.
The solution to this system of equations can be found using either elimination or substitution.
A. Solve this system by elimination since it is already in standard form and lined up nicely:
To solve by elimination, we need to eliminate one of the variables by adding or subtracting the equations. In this case, if we multiply the second equation by 2, we can eliminate the y variable.
The equations become:
8x - 2y = -4
8x - 2y = -4
Subtracting the second equation from the first equation eliminates the y variable:
(8x - 2y) - (8x - 2y) = -4 -(-2)
0 = -2
This result shows that the system of equations has no solution. The two equations represent parallel lines that will never intersect.
B. Solve this system by substitution since one of the variables is defined by the other without having to do any math:
In the second equation, we can solve for one variable in terms of the other. Rearranging the equation, we have:
4x - y = -2
y = 4x + 2
We can substitute this expression for y into the first equation:
8x - 2(4x + 2) = -4
Simplifying the equation gives:
8x - 8x - 4 = -4
-4 = -4
This equation is true, which means that any value of x would satisfy the equation. However, since we need a solution for both x and y, we can plug the expression for y back into the second equation:
4x - (4x + 2) = -2
Simplifying this equation gives:
4x - 4x - 2 = -2
-2 = -2
Again, this equation is true, which means any value of x would satisfy the equation. However, the y variable does not have a unique value; it depends on the value of x. Therefore, the system of equations has infinitely many solutions.
In summary, the solution to this system of equations is that there is no solution as the lines are parallel and do not intersect.
The solution to a system of equations can be found by using various methods. In this case, let's determine the most suitable method for solving the given system of equations:
8x - 2y = -4 (Equation 1)
4x - y = -2 (Equation 2)
To determine whether it would be better to solve this system by elimination or substitution, we should consider the structure of the equations.
In Equation 2, we see that y is already isolated on one side. This suggests that we could solve the system using substitution, as it simplifies the process.
Therefore, the correct answer is B. Solve this system by substitution since one of the variables is defined by the other without having to do any math.
To use substitution, you need to solve one equation for one variable in terms of the other variable. In this case, you can solve Equation 2 for y:
4x - y = -2
Rearranging the equation, we obtain:
y = 4x + 2
Now that we have an expression for y, we can substitute it into Equation 1:
8x - 2(4x + 2) = -4
Simplifying this equation gives us:
8x - 8x - 4 = -4
By eliminating like terms, the equation becomes:
-4 = -4
Since -4 is equal to -4, this equation is true. It means that the system of equations is dependent, meaning they represent the same line and infinitely many solutions exist.
Therefore, the solution to this system is:
x can take on any value, and y can be determined by substituting the value of x into the equation y = 4x + 2.