Reducing a System of Equations to One Variable

In summary, the system of equations presented involves three equations with two variables, but it is possible to obtain a single equation with only one variable by using substitution and solving for the unknown variable. This can be done by expressing x_2 in terms of x_1 and a, and then using this substitution in the other equations to eliminate x_2 and solve for x_1. The resulting equation will contain the cosine of an arcsine, but this can be simplified to obtain a final solution.
  • #1
st3dent
21
0
Hello, I am difficulty solving this system of equations.

Eqn1: 1.0416*10-21 = 4.75*10-27(x2) + 1.68*10-27(x1)(cos a)

Eqn2: 2.827 = (x1)(sin a)

Eqn3: 3.22896*10-16 = 8.4*10-28(x2)2 + 3.36*10-27(x1)2

I keep on getting equations with two variables in it. Can someone tell me how to get an eqn with only one variable out of this system.. Thanks!
 
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  • #2
st3dent said:
Hello, I am difficulty solving this system of equations.

Eqn1: 1.0416*10-21 = 4.75*10-27(x2) + 1.68*10-27(x1)(cos a)

Eqn2: 2.827 = (x1)(sin a)

Eqn3: 3.22896*10-16 = 8.4*10-28(x2)2 + 3.36*10-27(x1)2

I keep on getting equations with two variables in it. Can someone tell me how to get an eqn with only one variable out of this system.. Thanks!

Eq 3 let's you define [itex]x_2[/itex] in terms of [itex]x_1[/itex] as a simple ratio. Eq 2 let's you define a in terms of[itex]x_1[/itex] using an arcsin. Substituting these into Eq 1 will give you an equation with only one variable, although it will contain the cos of an arcsin.

Does that help?
 
  • #3
st3dent,

The cos(arcsin( )) in the solution that Bob3141592 showed you how to get can be simplified.
 
  • #4
Eq 2 gives you 'x_1 = 2.827/sin(a)'. Substitute this for x_1 in Eq 1 and 3, then in each solve for x_2 and equate. Can then calc 'a' and work back.
 
  • #5
solve the first equation for x_1 cos(a), square, then add to the second equation squared. Substitute into the third equation for x_1^2, you only have one unknown then...backsubstitute...
 
  • #6
Thank you.
 

FAQ: Reducing a System of Equations to One Variable

What is a system of equations?

A system of equations is a set of two or more equations that have a common set of variables. The solution to a system of equations is a set of values for the variables that makes all the equations true.

How do I solve a system of equations?

There are several methods for solving a system of equations, including substitution, elimination, and graphing. The most common method is substitution, where you solve for one variable in one equation and then substitute that value into the other equations. You continue this process until all variables have been solved for.

What is the importance of solving a system of equations?

Solving a system of equations is important because it allows us to find the values of variables that make multiple equations true. This is especially useful in real-world applications, such as calculating the intersection point of two lines or finding the optimal solution to a problem.

Can a system of equations have more than one solution?

Yes, a system of equations can have one, zero, or infinitely many solutions. A system with one solution is consistent and independent, meaning the equations intersect at one unique point. A system with no solutions is inconsistent, meaning the equations do not intersect. A system with infinitely many solutions is consistent and dependent, meaning the equations intersect at multiple points.

How do I check if my solution to a system of equations is correct?

You can check your solution by plugging the values into each equation and solving. If the solution is correct, all equations should be true. Additionally, you can graph the equations and see if the solution point lies on all the lines.

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