Systems of Equations by Graphs and Substitution, SAT math lesson

A school play sells 50 tickets and takes in 520 dollars. Adult tickets cost 12 dollars and student tickets cost 8 dollars. One equation counts tickets and another counts money, and neither one alone tells you how many adults came. Together they pin down a single answer. That is the idea of a system, two conditions that must hold at the same time.

See also Curve Intersections and Which Curve Is Higher in Foundations for Calculus and Linear Systems in Two and Three Variables in Precalculus.

Key terms

System of equations

Definition A system of equations is a set of two or more equations in the same variables that are meant to hold at the same time. A system of two linear equations in $x$ and $y$ has the form

\begin{aligned} a_1x + b_1y &= c_1 \\[0.4em] a_2x + b_2y &= c_2, \end{aligned}

where in each equation the coefficients of $x$ and $y$ are not both $0$ . The equations may also be written in other forms, such as $y = 2x + 1$ .

In plain words Two conditions on the same two unknowns. You are looking for the values that satisfy both of them at once.

Example For the ticket sale, let $a$ be the number of adult tickets and $s$ the number of student tickets. The system is

\begin{aligned} a + s &= 50 \\[0.4em] 12a + 8s &= 520. \end{aligned}

The first equation counts tickets and the second counts dollars.

Simultaneous solution

Definition A solution of a system in $x$ and $y$ is an ordered pair $(x, y)$ that makes every equation of the system true. The set of all such pairs is the solution set of the system. A pair that satisfies only some of the equations is not a solution. The order matters, since $(x, y)$ lists the value of $x$ first. When a question names the variables differently, follow the order it gives.

In plain words A solution has to pass every test, not just one of them. Report it with the $x$ -value first.

Example Consider $y = 2x + 1$ and $x + y = 7$ . The pair $(2, 5)$ is a solution, since $5 = 2(2) + 1$ and $2 + 5 = 7$ . The pair $(3, 4)$ satisfies $x + y = 7$ but not $y = 2x + 1$ , because $2(3) + 1 = 7 \ne 4$ . The pair $(5, 2)$ has the right numbers in the wrong order. It still satisfies $x + y = 7$ , but it fails $y = 2x + 1$ , so it is not a solution.

Intersection of graphs

Definition The graph of an equation in $x$ and $y$ is the set of all points $(x, y)$ that satisfy it. A point lies on both graphs exactly when it satisfies both equations, so the solutions of a system are exactly the intersection points of the graphs. For two lines in the plane there are three possibilities. They meet in one point, they are parallel and never meet, or they are the same line.

In plain words Draw both lines. Where they cross is the answer, because that point is on both lines at once.

Example Graph $y = 2x + 1$ and $y = -x + 5$ in the Desmos graphing calculator by typing each equation on its own line. Clicking the crossing point shows its coordinates as rounded decimals, such as $(1.333, 3.667)$ . A graph can guide you, but it cannot tell you whether $1.333$ is exactly $\frac{4}{3}$ or only close to it. Setting the right sides equal gives $2x + 1 = -x + 5$ , so $3x = 4$ , and the exact solution is

\left(\frac{4}{3}, \frac{11}{3}\right).

Substitution

Definition Substitution solves a system in four steps.

Solve one equation for one variable, for example $y$ in terms of $x$ .
Replace that variable in the other equation with the expression, which leaves an equation in one variable.
Solve that equation, then put the value back into the expression to get the other variable.
Check the pair in both original equations.

Replacing $y$ by $2x + 1$ is allowed because, for any solution, $y$ and $2x + 1$ are the same number. In terms of the graph, it means looking only at points on the line $y = 2x + 1$ and asking which of them also lies on the other line.

In plain words When one equation tells you what $y$ equals, use that to get rid of $y$ in the other equation. One equation with one unknown is something you already know how to solve.

Example Solve the system $y = 2x + 1$ and $3x + 2y = 16$ . Replace $y$ in the second equation.

\begin{aligned} 3x + 2(2x + 1) &= 16 \\[0.4em] 7x + 2 &= 16 \\[0.4em] x &= 2 \end{aligned}

Then $y = 2(2) + 1 = 5$ . Check both equations. The first gives $5 = 5$ and the second gives $6 + 10 = 16$ . The solution is $(2, 5)$ .

Choosing what to isolate

Definition If neither equation is already solved for a variable, solve one of them for a variable first. Any choice gives the same solution set, as long as you divide only by nonzero numbers. A variable with coefficient $1$ or $-1$ is the easiest choice, since isolating it creates no fractions.

In plain words Look for a variable that stands alone. Isolating it keeps the arithmetic clean.

Example Solve $x - 3y = 1$ and $2x + 5y = 13$ . The $x$ in the first equation has coefficient $1$ , so $x = 3y + 1$ . Substituting gives

2(3y + 1) + 5y = 13,

so $11y + 2 = 13$ and $y = 1$ . Then $x = 3(1) + 1 = 4$ . The check gives $4 - 3 = 1$ and $8 + 5 = 13$ , so the solution is $(4, 1)$ .

Solutions in context

Definition In a word problem, each variable stands for a named quantity with units. A solution of the system is meaningful only when its values make sense for those quantities, for example whole numbers for counts. The question decides which value to report, so read the last sentence again before answering.

In plain words Say in words what each number means. A question about student tickets wants the student count, not the adult count.

Example Back to the ticket system $a + s = 50$ and $12a + 8s = 520$ . From the first equation $s = 50 - a$ . Substituting,

12a + 8(50 - a) = 520,

so $4a + 400 = 520$ and $a = 30$ . Then $s = 20$ . The play sold 30 adult tickets and 20 student tickets, and the money checks, since $360 + 160 = 520$ .

Check yourself

Which ordered pair $(x, y)$ is the solution of the system $y = 3x - 2$ and $x + 2y = 17$ ? (A) $(3, 7)$ (B) $(7, 3)$ (C) $(5, 6)$ (D) $(1, 1)$

Show the answer

The answer is (A). Substituting gives $x + 2(3x - 2) = 17$ , so $7x = 21$ and $x = 3$ . Then $y = 9 - 2 = 7$ . Choice (B) lists the numbers in the wrong order. Choice (C) satisfies only the second equation, and choice (D) satisfies only the first, so neither solves the system.

In the $xy$ -plane, the graphs of $y = 2x$ and $x + y = 4$ intersect at the point $(a, b)$ . What is the value of $b$ ?

Show the answer

The answer is $\frac{8}{3}$ . Substituting $y = 2x$ gives $3x = 4$ , so $a = \frac{4}{3}$ and

b = 2 \cdot \frac{4}{3} = \frac{8}{3}.

Enter $8/3$ , or a decimal such as $2.666$ or $2.667$ . A rounded graph reading such as $2.667$ would also be accepted here, because it is $\frac{8}{3}$ rounded at the last digit that fits. The algebra is what tells you the exact value.

A club bought 40 notebooks for a total of 97 dollars. Some cost 2 dollars each and the rest cost 3 dollars each. How many 3-dollar notebooks did the club buy?

Show the answer

The answer is $17$ . Let $t$ be the number of 2-dollar notebooks and $h$ the number of 3-dollar notebooks. Then $t + h = 40$ and $2t + 3h = 97$ . With $t = 40 - h$ ,

2(40 - h) + 3h = 97,

so $80 + h = 97$ and $h = 17$ . The other $23$ notebooks cost 2 dollars, and $46 + 51 = 97$ checks. Entering $23$ answers the wrong question.

Systems of Equations by Graphs and Substitution

Theory, part 29

Practice, part 30

Key terms

System of equations

Simultaneous solution

Intersection of graphs

Substitution

Choosing what to isolate

Solutions in context

Check yourself

How this lesson connects

Builds on

Used later in