When given multiple linear equations and asked to solve for certain variables, the first requirement is that you have as many equations as you have variables. If you have more equations than variables, that's fine; just use whichever are the simplest. If you have more variables than equations, the best you can do is solve for one variable in terms of another.
In a scholastic setting, you would normally have the same number of variables as equations. That is what will be assumed for the remainder of this article. The most common format of a system of linear equations is this:
y=2x+3
y=x+5
When asked to solve these equations, what's really being asked is for you to find where those two lines intersect. This allows you to assume that the x value and y value will be the same at that point. Therefore, you can say that
y=2x+3=x+5
2x+3=x+5
x=2
Then, since we know that x=2, we can put that value into either of the initial equations to solve for y. Let's choose the second one, because then we don't have to worry about multiplying by two.
y=x+5
y=2+5
y=7
Therefore, the two lines intersect at the coordinates (2,7). This method is normally referred to as solving by comparison, which is a special case of solving by substitution (see below).
Sometimes, equations are not formatted in a way that solving by comparison is convenient. For example, perhaps something like this is given:
6=9y-3x
2y=3x+4
In situations like these, it is often necessary to do some algebra before combining the equations in some way. Let's try to isolate the variable x in the first equation.
6=9y-3x
3x+6=9y
3x=9y-6
x=3y-2
Now that we have x in terms of y, let's put that new equation into the second given equation.
2y=3(3y-2)+4
2y=9y-6+4
2y=9y-2
2=7y
y=2/7
Again, putting this y value into the first equation, we get:
6=9y-3x
6=9(2/7)-3x
6=18/7-3x
6-18/7=3x
42/7-18/7=3x (in this step, we multiplied 6 by 7/7)
24/7=3x
24/7*(1/3)=x
x=24/21
Therefore the coordinates that these two lines intersect at is (24/21,2/7).
You'll notice that this set of coordinates are less appealing than the integers found in the first problem. Since substitution is a very general method of solving linear equations, it will work under any circumstances. If you have doubts about using comparison (or the final method, outlined below) then use substitution. Provided you don't make any mistakes in the mechanics, substitution will get you the correct answer.
The final method is called elimination. Using this method, one can often avoid long and tedious algebra. For example:
2x+y=1
-2x+2y=5
If we were to use substitution, we would get the correct answer (though we'd have to isolate a variable, expand a bracket, isolate for the other variable in terms of the first, substitute that equation into the other, and eventually solve for both variables). However, there is an easier way. We can add the two equations together.
2x+(-2x)+y+2y=1+5
0+3y=6
y=2
Substituting y=2 into the first equation, we get:
2x+y=1
2x+2=1
2x=-1
x=-(1/2)
Therefore the coordinates that these two lines intersect at is (-1/2,2)
It is worth noting that we are not limited to adding the two equations together. We may also subtract, multiply, and divide the equations. Elimination often takes the most practice to spot when it will be useful, but it can save many lines of math when applied in a suitable manner (as in the last example).
Those are the basic methods for solving two equation, two unknown problems. They will also hold true for three or more equation/unknown problems, but the mechanics of the solution becomes very long, very quickly. In more than two equation/variable questions, it is recommended to use matrices, but that is beyond the scope of this answer.
You would solve them in exactly the same way as you would solve linear equations with real coefficients. Whether you use substitution or elimination for pairs of equations, or matrix algebra for systems of equations depends on your requirements. But the methods remain the same.
3(5x-2y)=18 5/2x-y=-1
Gaussian elimination is used to solve systems of linear equations.
Tell me the equations first.
You need as many equations as you have variables.
You can use a graph to solve systems of equations by plotting the two equations to see where they intersect
The answer depends on whether they are linear, non-linear, differential or other types of equations.
because you need maths in your life.. everyone does
solve systems of up to 29 simultaneous equations.
You would solve them in exactly the same way as you would solve linear equations with real coefficients. Whether you use substitution or elimination for pairs of equations, or matrix algebra for systems of equations depends on your requirements. But the methods remain the same.
A way to solve a system of equations by keeping track of the solutions of other systems of equations. See link for a more in depth answer.
Solving linear systems means to solve linear equations and inequalities. Then to graph it and describing it by statical statements.
Linear Algebra is a branch of mathematics that enables you to solve many linear equations at the same time. For example, if you had 15 lines (linear equations) and wanted to know if there was a point where they all intersected, you would use Linear Algebra to solve that question. Linear Algebra uses matrices to solve these large systems of equations.
7
3(5x-2y)=18 5/2x-y=-1
Gaussian elimination is used to solve systems of linear equations.
you apply the Laplace transform on both sides of both equations. You will then get a sytem of algebraic equations which you can solve them simultaneously by purely algebraic methods. Then take the inverse Laplace transform .