System of Equations

Sommaire

In short: System of equations refers to two or more equations sharing the same unknowns, and solving it means finding the values that make every one of them true at the same moment.

8th GradeAlgebra8.EE.C.8

What is System of Equations?

A system of equations bundles two conditions on the same unknowns. Mia has $\$15$ and saves $\$5$ a week; Leo has $\$45$ and saves $\$2$ a week. Each person gives one equation, and the question is when the two totals agree.

Graphed as two lines, the story is visible: a single crossing means one answer, parallel lines mean no answer, and two lines lying on top of each other mean endless answers. The crossing point $(x, y)$ satisfies both rules together.

How to spot it

A system of equations hides in any story that gives two separate rules for the same pair of quantities, like two savings plans.
The question asks when two amounts become equal, or which two numbers meet two conditions.
Two lines are drawn on one graph and the intersection point is requested.
Equations come in pairs, each with the same variables x and y.

The method

Build the system of equations by writing one equation per condition, with shared variables.
Solve with graphing, substitution or elimination; choose whichever fits the form given.
Find the second variable by putting the first value back into either equation.
Check the pair in both original equations, then answer in words with units.

The lines meet at (10, 65): after 10 weeks both have $65.

System of Equations: solved example

Problem

Mia has $\$15$ and adds $\$5$ each week. Leo has $\$45$ and adds $\$2$ each week. Write a system and find when their savings are equal.

Solution

Let $w$ count weeks and $y$ count dollars saved: $y = 5w + 15$ for Mia and $y = 2w + 45$ for Leo.

Equal totals mean $5w + 15 = 2w + 45$, so $3w = 30$ and $w = 10$. Then $y = 5(10) + 15 = 65$. Both reach $\$65$ at the 10-week mark.

Try it

Question 1

Solve the system $y = x + 3$ and $y = -2x + 9$.

Show the answer

$x + 3 = -2x + 9$ gives $3x = 6$, so $x = 2$ and $y = 5$: the solution is $(2, 5)$.

Question 2

Count the solutions when $y = 2x + 1$ is paired with $y = 2x – 4$.

Show the answer

Equal slopes and different intercepts make the lines parallel, so they never meet and no solution exists.

System of Equations vs. Linear Equation

A single linear equation in two variables has infinitely many solutions, one for every point on its line. A second condition usually narrows those possibilities to the one point shared by both lines, so the answer is an ordered pair rather than a lone number.

Linear Equation

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Whole dictionary: A–Z · 8th Grade · Algebra

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