Linear equations, inequalities and systems

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1. Solve linear equations in one variable

Warm-up

Three quick questions from earlier lessons.

Discover

A pizza party costs 40 dollars plus 12 dollars per pizza, and the total is 100 dollars. Which equation fits?

$40 + 12p = 100$
$12 + 40p = 100$

Key idea

Equivalent equations have the same solutions.

To solve a linear equation, make equivalent equations: add or subtract the same amount on both sides, multiply or divide both sides by the same non-zero number, distribute, collect like terms. Stop when

x

is alone.

Worked example

Solve $5x + 3 = 2x + 15$ .

3x + 3 = 15

Subtract 2x.

3x = 12

Subtract 3.

x = 4

Divide by 3.

Your turn, guided

Your turn: $3(x + 2) = 21$ . Find $x$ .

x = 5

Divide by 3: x + 2 = 7.

Practice ★

Solve $3x + 3 = -18$ . $x = \;?$
Solve $8x + 14 = -26$ . $x = \;?$

Solve $4x + 8 = 3x + 9$ . $x = \;?$

Watch out!

Practice ★★

Solve $6(x + 4) = 30$ . $x = \;?$
Solve $3(x + 4) = 33$ . $x = \;?$

Exit ticket

Three questions to check you have got it.

Solve $3x + 3 = -18$ . $x = \;?$
Solve $8x + 14 = -26$ . $x = \;?$
Solve $4x + 4 = 32$ . $x = \;?$

Summary card

Linear equations

Do the same to both sides until $x$ is alone.

Watch out: distribute to every term.

2. Solve a formula for a variable

Warm-up

Three quick questions from earlier lessons.

Discover

A rectangle has perimeter $P = 2\ell + 2w$ . You know $P$ and $\ell$ . Which formula gives $w$ directly?

$w = \dfrac{P - 2\ell}{2}$
$w = P - \dfrac{2\ell}{2}$

Key idea

Solve for a letter as you solve for $x$ .

To solve for a variable, treat the other letters as numbers: use the same moves as for a number equation until that variable is alone. Divide every term on the other side.

Worked example

Solve $3x + 2y = 12$ for $y$ .

2y = 12 - 3x

Subtract 3x.

y = \dfrac{12 - 3x}{2}

Divide both terms by 2.

Your turn, guided

Your turn: $y = 4x + 8$ . Solve for $x$ .

x = \dfrac{y - 8}{4}

Subtract 8, divide by 4.

Practice ★

A car moves at 2 miles per minute, so $d = 2t$ . Solve for $t$ . $t = \;?$
A car moves at 4 miles per minute, so $d = 4t$ . Solve for $t$ . $t = \;?$

Solve $7x + 9y = 23$ for $y$ . $y = \;?$

Watch out!

Practice ★★

Solve $y = 9x + 3$ for $x$ . $x = \;?$
Solve $y = 2x + 4$ for $x$ . $x = \;?$

Exit ticket

Three questions to check you have got it.

Solve $7x + 9y = 23$ for $y$ . $y = \;?$
Solve $7x + 5y = 13$ for $y$ . $y = \;?$
Solve $7x + 8y = 7$ for $y$ . $y = \;?$

Summary card

Solving for a variable

Same moves as with numbers; the other letters are just numbers you do not know.

Watch out: divide every term.

3. Linear equations in two variables and their graphs

Warm-up

Three quick questions from earlier lessons.

Discover

Is $(2, 4)$ a solution of $3x + y = 10$ ?

Key idea

A linear equation in two variables is a line.

The solutions of

Ax + By = C

are the points of a line. Set

y = 0

for the $x$ -intercept and

x = 0

for the $y$ -intercept. Solving for

y

gives

y = mx + b

, with slope

m = -\tfrac{A}{B}

Worked example

$2x + 5y = 20$ .

x

-intercept

10

y = 0.

y

-intercept

4

x = 0.

Slope

-\tfrac{2}{5}

y = −(2/5)x + 4.

Your turn, guided

Your turn: the $y$ -intercept of $3x + 4y = 24$ .

6

x = 0: 4y = 24.

Practice ★

The point $(2, y)$ is a solution of $5x + y = 15$ . Find $y$ .
The point $(1, y)$ is a solution of $5x + y = 17$ . Find $y$ .

Find the $x$ -intercept of $3x + 6y = 54$ (the value of $x$ where the line crosses the $x$ -axis).

Watch out!

Practice ★★

What is the slope of the line $8x + 9y = 17$ ?
What is the slope of the line $1x + 4y = 4$ ?

Exit ticket

Three questions to check you have got it.

Find the $x$ -intercept of $3x + 6y = 54$ (the value of $x$ where the line crosses the $x$ -axis).
Find the $x$ -intercept of $3x + 1y = 3$ (the value of $x$ where the line crosses the $x$ -axis).
Find the $x$ -intercept of $4x + 1y = 12$ (the value of $x$ where the line crosses the $x$ -axis).

Summary card

Lines from equations

Intercepts: set the other variable to 0. Slope: solve for $y$ .

Watch out: slope of $Ax + By = C$ is $-A/B$ .

4. Solve systems of linear equations by substitution

Warm-up

Three quick questions from earlier lessons.

Discover

$y = 2x$ and $x + y = 9$ . Since $y$ is $2x$ , you can replace it. What is $x$ ?

Key idea

Substitution turns two equations into one.

A solution of a system makes both equations true: it is where the lines cross. Substitution: solve one equation for a variable, put that expression into the other equation, solve, then substitute back to get the other variable.

Worked example

$y = 3x - 1$ and $2x + y = 14$ .

2x + 3x - 1 = 14

Replace y.

x = 3

5x = 15.

y = 8

Back into y = 3x − 1.

Your turn, guided

Your turn: $y = x + 4$ and $x + y = 10$ . Find $x$ .

x = 3

2x + 4 = 10.

Practice ★

Solve the system $y = 2x + 4$ and $3x + y = 24$ . $x = \;?$
Solve the system $y = 5x + 5$ and $3x + y = 37$ . $x = \;?$

Solve the system $y = 2x + 4$ and $2x + y = 24$ . $y = \;?$

Watch out!

Practice ★★

Solve the system $x = 3y$ and $x + y = 24$ . $y = \;?$
Solve the system $x = 5y$ and $x + y = 24$ . $y = \;?$

Exit ticket

Three questions to check you have got it.

Solve the system $y = 2x + 4$ and $3x + y = 24$ . $x = \;?$
Solve the system $y = 5x + 5$ and $3x + y = 37$ . $x = \;?$
Solve the system $y = 5x + 7$ and $3x + y = 39$ . $x = \;?$

Summary card

Substitution

Replace one variable by its expression, solve, then substitute back.

Watch out: answer the variable the question asks for.

5. Solve systems of linear equations by elimination

Warm-up

Three quick questions from earlier lessons.

Discover

$x + y = 10$ and $x - y = 4$ . Add the two equations. What happens?

$y$ disappears: $2x = 14$
$x$ disappears

Key idea

Elimination: add or subtract whole equations.

Adding or subtracting equations keeps the solution of the system. Elimination: make one variable's coefficients equal (multiply an equation if needed), then subtract (or add if they are opposite) so that variable disappears.

Worked example

$2x + 5y = 19$ and $2x + y = 11$ .

4y = 8

Subtract: 2x cancels.

y = 2

Divide.

x = 4.5

2x + 2 = 11.

Your turn, guided

Your turn: $x + y = 12$ and $x - y = 2$ . Find $x$ .

x = 7

Add: 2x = 14.

Practice ★

Solve the system $x + y = 15$ and $x - y = 3$ . $x = \;?$
Solve the system $x + y = 3$ and $x - y = -1$ . $x = \;?$

Solve the system $3x + 4y = 30$ and $3x + 2y = 18$ . $y = \;?$

Watch out!

Practice ★★

To solve $x + 7y = 20$ and $9x + y = 20$ by elimination, you multiply the first equation by a number and subtract the second, so that $x$ disappears. Which number?
To solve $x + 5y = 20$ and $9x + y = 27$ by elimination, you multiply the first equation by a number and subtract the second, so that $x$ disappears. Which number?

Boss problem. A school play sold 33 tickets: adult tickets at 9 dollars and child tickets at 7 dollars, for 287 dollars in all. Let $x$ be the number of adult tickets and $y$ the number of child tickets.

Exit ticket

Three questions to check you have got it.

Solve the system $x + y = 15$ and $x - y = 3$ . $x = \;?$
Solve the system $x + y = 3$ and $x - y = -1$ . $x = \;?$
Solve the system $x + y = 8$ and $x - y = -6$ . $x = \;?$

Summary card

Elimination

Equal coefficients: subtract. Opposite coefficients: add.

Watch out: multiply first when no coefficients match.

6. Systems with one, no or infinitely many solutions

Warm-up

Three quick questions from earlier lessons.

Discover

$y = 2x + 1$ and $y = 2x + 5$ . Can both be true at once?

No, never
Yes, always

Key idea

Two lines meet once, never, or everywhere.

Write both equations as

y = mx + b

. Different slopes: one solution. Same slope, different intercepts: parallel lines, no solution. Same slope and intercept: the same line, infinitely many solutions.

Worked example

$y = 3x - 2$ and $6x - 2y = 4$ .

y = 3x - 2

Solve the second for y.

Infinitely many solutions

Identical.

Your turn, guided

Your turn: how many solutions has $y = 4x + 1$ , $y = -x + 6$ ? Type the number.

1

Different slopes.

Practice ★

How many solutions has the system $y = 5x + 6$ and $y = 5x + 7$ ?
1. No solution
2. Exactly one
3. Infinitely many
How many solutions has the system $y = 3x + 9$ and $y = 3x + 11$ ?
1. No solution
2. Exactly one
3. Infinitely many

How many solutions has the system $y = 3x + 2$ and $y = -3x + 9$ ?
1. Exactly one
2. No solution
3. Infinitely many

Watch out!

Practice ★★

How many solutions has the system $y = 6x + 4$ and $12x - 2y = -8$ ?
1. Infinitely many
2. No solution
3. Exactly one
How many solutions has the system $y = 5x + 4$ and $10x - 2y = -8$ ?
1. Infinitely many
2. No solution
3. Exactly one

Exit ticket

Three questions to check you have got it.

How many solutions has the system $y = 5x + 6$ and $y = 5x + 7$ ?
1. No solution
2. Exactly one
3. Infinitely many
How many solutions has the system $y = 3x + 9$ and $y = 3x + 11$ ?
1. No solution
2. Exactly one
3. Infinitely many
How many solutions has the system $y = 4x + 8$ and $y = 4x + 10$ ?
1. No solution
2. Exactly one
3. Infinitely many

Summary card

How many solutions?

Different slopes: one. Parallel: none. Same line: infinitely many.

Watch out: same slope alone is not enough; check the intercepts.

7. Solve linear inequalities in one variable

Warm-up

Three quick questions from earlier lessons.

Discover

$-2x > 6$ . Try $x = -4$ : $-2 \cdot (-4) = 8 > 6$ . So which describes all solutions?

$x < -3$
$x > -3$

Key idea

Solve inequalities like equations, with one extra rule.

Adding, subtracting, multiplying or dividing by a positive number keeps the inequality. Multiplying or dividing by a negative number flips it. Strict signs (

<

>

) exclude the endpoint;

\le

\ge

include it.

Worked example

Solve $-3x + 4 \le 13$ .

-3x \le 9

Subtract 4.

x \ge -3

Divide by −3: flip.

[-3, \infty)

Endpoint included.

Your turn, guided

Your turn: $2x - 1 < 9$ . The boundary value of $x$ .

x < 5

2x < 10.

Practice ★

Solve $8x + 5 < 45$ . Give the solution set as an interval.
Solve $6x + 5 < 59$ . Give the solution set as an interval.

You have 109 dollars and tickets cost 9 dollars each. Solve $9t \le 109$ : what is the greatest number of tickets you can buy?

Watch out!

Practice ★★

Solve $-2x + 17 \le 9$ . Give the solution set as an interval.
Solve $-2x + 1 \le 19$ . Give the solution set as an interval.

Exit ticket

Three questions to check you have got it.

Solve $8x + 5 < 45$ . Give the solution set as an interval.
Solve $6x + 5 < 59$ . Give the solution set as an interval.
Solve $7x + 9 < -54$ . Give the solution set as an interval.

Summary card

Inequalities

Solve as an equation; flip when multiplying or dividing by a negative.

Watch out: check with a number from your answer.

8. Absolute value equations and inequalities

Warm-up

Three quick questions from earlier lessons.

Discover

Which numbers are at a distance of 3 from 0 on the number line?

3 and −3
Only 3

Key idea

Absolute value is distance.

|x - a|

is the distance between

x

and

a

. So

|x - a| = b

(with

b > 0

) has two solutions,

a - b

and

a + b

;

|x - a| < b

is the interval

a - b < x < a + b

;

|x - a| > b

is the two outer pieces

x < a - b

x > a + b

. If

b < 0

|x - a| = b

has no solution.

Worked example

Solve $|x - 2| = 5$ and $|x - 2| \le 5$ .

x - 2 = 5

x - 2 = -5

x = 7

x = -3

Two cases.

-3 \le x \le 7

Inequality.

Your turn, guided

Your turn: the larger solution of $|x - 4| = 6$ .

10

4 + 6.

Practice ★

Solve $|x - (4)| = 4$ . Enter all solutions.
Solve $|x - (-6)| = 5$ . Enter all solutions.

How many solutions does the equation |x + 9| = −7 have?
1. None
2. Two
3. One: x = -2

Watch out!

Practice ★★

Solve $|x - (4)| < 4$ . Give the solution set as an interval.
Solve $|x - (9)| < 6$ . Give the solution set as an interval.

Solve $|x - (2)| \geq 2$ . Give the solution set as an interval or a union of intervals.

Exit ticket

Three questions to check you have got it.

Solve $|x - (4)| = 4$ . Enter all solutions.
Solve $|x - (-6)| = 5$ . Enter all solutions.
Solve $|x - (4)| = 3$ . Enter all solutions.

Summary card

Absolute value equations and inequalities

|x − a| = b: a ± b. Less than: between. Greater than: outside.

Watch out: do not forget the negative case.

9. Linear inequalities and systems of inequalities in two variables

Warm-up

Three quick questions from earlier lessons.

Discover

Is the point $(0, 0)$ a solution of $y > 2x + 3$ ?

Key idea

An inequality in two variables is a half-plane.

Graph

y < mx + b

y > mx + b

with a dashed boundary line, and

\le

\ge

with a solid line. Shade above for

y >

and below for

y <

(or test a point). The solutions of a system of inequalities are where the shadings overlap.

Worked example

$y \ge -x + 4$ .

Solid line

y = -x + 4

≥ includes the line.

Shade the side without the origin: above

Test (0, 0): 0 ≥ 4 is false.

Your turn, guided

Your turn: is $(1, 6)$ a solution of $y > 3x + 2$ ? Type 1 for yes, 0 for no.

1

6 > 5: yes.

Practice ★

Which point is a solution of $y > 3x + 3$ ?
1. $(0, 4)$
2. $(0, 3)$
3. $(0, 2)$
Which point is a solution of $y > 5x + 7$ ?
1. $(0, 8)$
2. $(0, 7)$
3. $(0, 6)$

How do you graph $y \le 6x + 2$ ?
1. Solid line, shade below
2. Dashed line, shade below
3. Solid line, shade above

Watch out!

Practice ★★

Adult tickets cost 15 dollars and child tickets 5 dollars. A group can spend at most 168 dollars on $x$ adult and $y$ child tickets. Which inequality models this?
1. $15x + 5y \le 168$
2. $15x + 5y \ge 168$
3. $15x + 5y < 168$
Adult tickets cost 9 dollars and child tickets 3 dollars. A group can spend at most 126 dollars on $x$ adult and $y$ child tickets. Which inequality models this?
1. $9x + 3y \le 126$
2. $9x + 3y \ge 126$
3. $9x + 3y < 126$

Exit ticket

Three questions to check you have got it.

Which point is a solution of $y > 3x + 3$ ?
1. $(0, 4)$
2. $(0, 3)$
3. $(0, 2)$
Which point is a solution of $y > 5x + 7$ ?
1. $(0, 8)$
2. $(0, 7)$
3. $(0, 6)$
Which point is a solution of $y > 3x + 8$ ?
1. $(0, 9)$
2. $(0, 8)$
3. $(0, 7)$

Summary card

Inequalities in two variables

Dashed for $<, >$ ; solid for $\le, \ge$ . Test a point to choose the side.

Watch out: “at most” is $\le$ , “at least” is $\ge$ .

Answers: Linear equations, inequalities and systems

1. Solve linear equations in one variable

$-7$
$-5$

$-7$
$-5$
$7$

2. Solve a formula for a variable

$\frac{d}{2}$
$\frac{d}{4}$

$\frac{23}{9} - \frac{7 x}{9}$

$\frac{y}{9} - \frac{1}{3}$
$\frac{y}{2} - 2$

$\frac{23}{9} - \frac{7 x}{9}$
$\frac{13}{5} - \frac{7 x}{5}$
$\frac{7}{8} - \frac{7 x}{8}$

3. Linear equations in two variables and their graphs

$5$
$12$

$18$

$- \frac{8}{9}$
$- \frac{1}{4}$

$18$
$1$
$3$

4. Solve systems of linear equations by substitution

$14$

5. Solve systems of linear equations by elimination

6. Systems with one, no or infinitely many solutions

No solution
No solution

Exactly one

Infinitely many
Infinitely many

No solution
No solution
No solution

7. Solve linear inequalities in one variable

$(-\infty, 5)$
$(-\infty, 9)$

$12$

$[4, \infty)$
$[-9, \infty)$

$(-\infty, 5)$
$(-\infty, 9)$
$(-\infty, -9)$

8. Absolute value equations and inequalities

$0,\ 8$
$-11,\ -1$

None

$(0, 8)$
$(3, 15)$

$(-\infty, 0] \cup [4, \infty)$

$0,\ 8$
$-11,\ -1$
$1,\ 7$

9. Linear inequalities and systems of inequalities in two variables

$(0, 4)$
$(0, 8)$

Solid line, shade below

$15x + 5y \le 168$
$9x + 3y \le 126$

$(0, 4)$
$(0, 8)$
$(0, 9)$