There are always three ways to solve a system of equations
There are three ways to solve systems of linear equations: substitution, elimination, and graphing. Let’s review the steps for each method.
Substitution
Get a variable by itself in one of the equations.
Take the expression you got for the variable in step 1, and plug it (substitute it using parentheses) into the other equation.
Solve the equation in step 2 for the remaining variable.
Use the result from step 3 and plug it into the equation from step 1.
Elimination
If necessary, rearrange both equations so that the ???x???-terms are first, followed by the ???y???-terms, the equals sign, and the constant term (in that order). If an equation appears to have not constant term, that means that the constant term is ???0???.
Multiply one (or both) equations by a constant that will allow either the ???x???-terms or the ???y???-terms to cancel when the equations are added or subtracted (when their left sides and their right sides are added separately, or when their left sides and their right sides are subtracted separately).
Add or subtract the equations.
Solve for the remaining variable.
Plug the result of step 4 into one of the original equations and solve for the other variable.
Graphing
Solve for ???y???in each equation.
Graphboth equations on the same Cartesian coordinate system.
Find the point of intersection point of the lines (the point where the lines cross).
The easiest way to solve this system would be to use substitution since ???x???is already isolated in the first equation. Whenever one equation is already solved for a variable, substitution will be the quickest and easiest method.
Even though you’re not asked to solve, these are the steps to solve the system:
Substitute ???y+2???for ???x???in the second equation.
???3y-2(y+2)=15???
Distribute the ???-2???and then combine like terms.
???3y-2y-4=15???
???y-4=15???
Add ???4???to both sides.
???y-4+4=15+4???
???y=19???
Plug ???19??? for ???y???intothe first equation.
???x=y+2???
???x=19+2???
???x=21???
The unique solution is ???(21,19)???.
How to solve a system using the elimination method
Example
To solve the system by elimination, what would be a useful first step?
???x+3y=12???
???2x-y=5???
When we use elimination to solve a system, it means that we’re going to get rid of (eliminate) one of the variables. So we need to be able to add the equations, or subtract one from the other, and in doing so cancel either the ???x???-terms or the ???y???-terms.
Any of the following options would be a useful first step:
Multiply the first equation by ???-2???or ???2???. This would give us ???2x???or ???-2x???in both equations, which will cause the ???x???-terms to cancel when we add or subtract.
Multiply the second equation by ???3???or ???-3???. This would give us ???3y???or ???-3y???in both equations, which will cause the ???y???-terms to cancel when we add or subtract.
Divide the second equation by ???2???. This would give us ???x???or ???-x???in both equations, which will cause the ???x???-terms to cancel when we add or subtract.
Divide the first equation by ???3???. This would give us ???y???or ???-y???in both equations, which will cause the ???y???-terms to cancel when we add or subtract.
Let’s re-do the last example, but instead of the elimination method, use a graph to find the solution.
Solving the system by graphing both equations and finding the intersection points
Example
Graph both equations to find the solution to the system.
???x+3y=12???
???2x-y=5???
In order to graph these equations, let’s put both of them into slope-intercept form. We get
???x+3y=12???
???3y=-x+12???
???y=-\frac13x+4???
and
???2x-y=5???
???-y=-2x+5???
???y=2x-5???
The line ???y=-(1/3)x+4??? intersects the ???y???-axis at ???4???, and then has a slope of ???-1/3???, so its graph is
The line ???y=2x-5??? intersects the ???y???-axis at ???-5???, and then has a slope of ???2???, so if you add its graph to the graph of ???y=-(1/3)x+4???, you get
Looking at the intersection point, it appears as though the solution is approximately ???(3.75,2.75)???. In actuality, the solution is ???(27/7,19/7)\approx(3.86,2.71)???, so our visual estimate of ???(3.75,2.75)??? wasn’t that far off.
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So, in order to solve that problem, you need to be able to find the value of all the variables in each equation. There are three different ways that you could do this: the substitution method, elimination method, and using an augmented matrix.
The Matrix method is the easiest way to solve a set of linear equations, because it is straightforward and a step-by-step method, and it boils down to the same thing as the elimination method that most people are familiar with.
Pick any two pairs of equations from the system. Eliminate the same variable from each pair using the Addition/Subtraction method. Solve the system of the two new equations using the Addition/Subtraction method. Substitute the solution back into one of the original equations and solve for the third variable.
The goal of the substitution method is to rewrite one of the equations in terms of a single variable. Equation B tells us that x=y+5, so it makes sense to substitute that y+5 into Equation A for x. Substitute y+5 into Equation A for x and you get y+(y+5)=3.
What is elimination with examples? 3x + y = 4 and -3x + y = -2 is considered a system of equations. Adding these two equations together will result in the elimination of the x variable. This means that the solution for y can be found and substituted back into the equation to find the value of x.
In this method, we multiply both the equations with a non-zero number to make the coefficients of any one variable equal. Then we add or subtract the equations to eliminate one of the variables to find the value of the other variable. This is how we solve linear equations by the elimination method.
There are three ways to solve a system of linear equations: graphing, substitution, and elimination. The solution to a system of linear equations is the ordered pair (or pairs) that satisfies all equations in the system. The solution is the ordered pair(s) common to all lines in the system when the lines are graphed.
A system of equations as discussed above is a set of equations that seek a common solution for the variables included. The following set of linear equations is an example of the system of equations: 2x - y = 12. x - 2y = 48.
These methods include: Newton's method, Broyden's method, and the Finite Difference method. where xi → x (as i → ∞), and x is the approximation to a root of the function f(x).
There are three methods by which simultaneous equations can be solved: elimination method, substitution method, graphing method. No matter which method is used, each method will lead to the same answer; however, there are times when one method leads to simpler calculations.
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