Introduction to Systems of Equations
Define a system of equations and its solution as an ordered pair that satisfies both equations simultaneously. Understand a system's solution graphically as the intersection point of two lines, and recognize consistent, inconsistent, and dependent systems.
A lesson of Algebra 1, in Systems of Equations & Inequalities, about 25 minutes.
What a System Is
By the end of this lesson, you will be able to say what a system of equations is, check whether a pair of numbers solves one, and explain why the solution is where the two lines cross.
Two Conditions at Once
A single linear equation in two variables has infinitely many solutions.
$$x + y = 10$$
$(1, 9)$ works. So does $(4, 6)$, $(0, 10)$, $(-3, 13)$, and every other pair adding to ten. Drawn, that is a whole line of them.
A system puts two such conditions on the same two unknowns and asks which pair satisfies both:
$$\begin{cases} x + y = 10 \ x - y = 2 \end{cases}$$
Now most of that infinite list is gone. $(4, 6)$ satisfies the first and fails the second. Only $(6, 4)$ does both, and that pair is the solution.
The whole idea in one line: each equation narrows the possibilities to a line, and asking for both at once narrows it to where those lines meet.
A Solution Satisfies Every Equation
This is the definition, and it is also the check.
$$\begin{cases} 2x + y = 11 \ x - y = 1 \end{cases}$$
Is $(4, 3)$ a solution?
$$2(4) + 3 = 11 \quad \checkmark \qquad 4 - 3 = 1 \quad \checkmark$$
Both hold, so yes.
Is $(5, 1)$ a solution?
$$2(5) + 1 = 11 \quad \checkmark \qquad 5 - 1 = 4 \neq 1 \quad \times$$
One out of two is not a solution. Both, or it is not one, and the most common error in the whole unit is checking the equation that was easiest and stopping there.
Example 1: Checking a Candidate
Here is the whole method on one pair, worked all the way through. $$\begin{cases} 3x + 2y = 16 \ x - y = 3 \end{cases}$$ Is $(4, 2)$ a solution?
- Substitute into the first equation. What does the left side come to? 16
- Does the first equation hold? Type yes or no yes
- Now the second. What does the left side come to? 2
- The second equation needs the left side to be 3. By how much is it short? 1
- So is (4, 2) a solution to the system? Type yes or no no
Worked Example 1: Your Turn
$$\begin{cases} x + 3y = 14 \ 2x - y = 7 \end{cases}$$ Is $(5, 3)$ a solution?
Sort It: Solution or Not
For the system $x + y = 8$ and $x - y = 2$, which candidates work?
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