How to Solve System of Two Linear Equations
- Thread starter Jason76
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How to solve the system below?What variable are we solving for?
System (a)
\(\displaystyle 2x_{1} - 4x_{2} = 6\)
\(\displaystyle 3x_{1} = 9\)
You;re looking for both x1 and x2.
Addition method: Add three times the first to two times the second to solve for x2.
Substitution method (easier in this case): Solve equation (2) for x1, and substitute the value into equation(1).
Yes for this situation, but no generally.Ok i get it now. We solve the equation which can be solved, the one with only one variable, then we plug that value into the other equation. The goal being to find the value of the other variable.
If we have two equations with two unknowns, we first use one equation to solve for one unknown in terms of the other unknown (not a numeric solution). Then we reduce the other equation to one unknown by substituting for the unknown that is known in terms of the other unknown. Then we get numeric solutions for the remaining unknown. And finally, we get a numeric solution for the first unknown. This is abstract so here is an example.
\(\displaystyle x^2 - 2xy + y^2 = 81.\)
\(\displaystyle x + y = 5.\)
Step 1: Solve for one unknown in terms of the other.
You can use either equation to do so. In this case, it is obviously easier to use the second equation.
\(\displaystyle y = 5 - x.\)
Step 2: Reduce the other equation to an equation in one unknown by substitution.
\(\displaystyle x^2 - 2xy + y^2 = 81 \implies x^2 - 2x(5 - x) + (5 - x)^2 = 81 \implies x^2 - 10x + 2x^2 + 25 - 10x + x^2 = 81 \implies 4x^2 - 20x - 56 = 0.\)
Step 3: Solve the reduced equation numerically.
\(\displaystyle 4x^2 - 20x - 56 \implies x^2 - 5x - 14 = 0 \implies (x - 7)(x + 2) = 0 \implies x = 7\ or\ x = - 2.\)
Step 4: Use the other equation to solve for the other unknown numerically.
\(\displaystyle x = 7 \implies 7 + y = 5 \implies y = - 2.\)
\(\displaystyle x = - 2 \implies - 2 + y = 5 \implies y = 7.\)
So the solutions are (7, - 2) and (- 2, 7).
We helped you solve systems of linear equations in 2012 and 2013.
Do you recall any of the following?
coefficient matrix
augmented matrix
matrix equation
row operations
reduced echelon form (ref)
row reduced echelon form (rref)
elimination method
substitution method
Cramer's Rule
:cool: