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1. 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.

2. 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.

3. 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.

4. 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: Systems of equations and inequalities

1. Solve systems of linear equations by substitution

$14$

2. Solve systems of linear equations by elimination

3. Systems with one, no or infinitely many solutions

No solution
No solution

Exactly one

Infinitely many
Infinitely many

No solution
No solution
No solution

4. Linear inequalities and systems of inequalities in two variables

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

Solid line, shade below

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

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