You are solving two equations, so two curves are involved it can be more, of course, but for simplicity let's stick with two curves. You are looking for points which are on both curves lines in your example. The only way a point can be on both curves is if the curves intersect at those points.
Any other points on either line will not be on both lines will not satisfy both equations at the same time. You should verify this for yourself—the proof is fairly obvious. In the case of a pair of linear equations, the solution to the system is the intersection of the two lines represented by the equations. This interpretation generalizes to more linear equations with more variables, but instead of lines, each individual equation represents a hyperplane.
Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Math Algebra all content System of equations Equivalent systems of equations and the elimination method. Systems of equations with elimination: King's cupcakes.
Practice: Systems of equations with elimination. Systems of equations with elimination: potato chips. Systems of equations with elimination and manipulation. The following are two more examples showing how to solve linear systems of equations using elimination. Notice the coefficients of each variable in each equation. If you add these two equations, the x term will be eliminated since. Add and solve for y. The solution is 5, 3. Use elimination to solve for x and y.
Change one of the equations to its opposite, add and solve for x. The solution is 2, 3. Go ahead and check this last example—substitute 2, 3 into both equations. Notice that you could have used the opposite of the first equation rather than the second equation and gotten the same result.
Using Multiplication and Addition to Eliminate a Variables. Many times adding the equations or adding the opposite of one of the equations will not result in eliminating a variable.
Look at the system below. If you add the equations above, or add the opposite of one of the equations, you will get an equation that still has two variables. You can multiply both sides of one of the equations by a number that will result in the coefficient of one of the variables being the opposite of the same variable in the other equation. This is where multiplication comes in handy. Notice that the first equation contains the term 4 y , and the second equation contains the term y.
Solve for x and y. Look for terms that can be eliminated. The equations do not have any x or y terms with the same coefficients. Rewrite the system, and add the equations. The solution is 4, There are other ways to solve this system. Instead of multiplying one equation in order to eliminate a variable when the equations were added, you could have multiplied both equations by different numbers. The equations do not have any x or y terms with the same coefficient.
In order to use the elimination method, you have to create variables that have the same coefficient—then you can eliminate them. When the coefficients of one variable are opposites you add the equations to eliminate a variable and when the coefficients of one variable are equal you subtract the equations to eliminate a variable.
To avoid errors make sure that all like terms and equal signs are in the same columns before beginning the elimination.
If you don't have equations where you can eliminate a variable by addition or subtraction you directly you can begin by multiplying one or both of the equations with a constant to obtain an equivalent linear system where you can eliminate one of the variables by addition or subtraction.
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