Multi-Step Linear Equations with number and type of solutions

Solving Equations with Variables on Both Sides
(Continued)

In your notes, solve the following equation.

4x - 7 = -3x

​

​Warm up

Multiple Choice

Identify the correct first step in solving this question:

-3(w + 4) = 4w - 5

• Add 5 to both sides

• Divide both sides by -3.

• Distribute -3 to the terms inside parentheses

• Subtract 4w from both sides

Solve the following question

In your notes, solve the following question:

​-3(w + 4) = 4w - 5

Solve the following question for the variable w.

​-3(w + 4) = 4w - 5
-3w - 12 = 4w -5
+3w +3w
- 12 = 7w - 5
+ 5 + 5
-7 = 7w
7 7
-1 = w

In this lesson today,

Students will be able to:

• Identify the number and type solutions for linear equations with variables on both sides.

When solving equations with variables on both sides,

•  The goal is to get the variable on one side (isolate the variable) and the constant on the other side.

• ** Strategy * Always move the smallest coefficient first to keep the variable term positive.

Multiple Choice

Identify the best possible first step to solve this equation:

g - 10 + 7g = 15 + 3g

• Combine like terms

• Subtract 10 from both sides

• Subtract g and 7g from each other

• Divide both sides by 3

Multiple Choice

Solve the following equation for g:

g - 10 + 7g = 15 + 3g

• g = 10

• g = 5

• g = -5

• g = 11/5

Linear Equations with Special Solutions

​Equations do not ALWAYS have just one solution. Linear equations may have the following results:

1) One Solution – The coefficient on sides of the equal sign is different. This means you have two lines that intersect at that point

2) No Solution – Final statement is NOT TRUE. ex. 5 = 0 This means your two lines are parallel to each other. **Tip** Coefficients on each side will be the same while the constants are different.

3) Infinitely Many Solutions (Identity) or Infinite Solutions – Final statement is TRUE. Ex. 0 = 0 or
-5 = -5. This means the lines are co-insiding lines (or the same lines) **Tip** Coefficients and constants are exactly the same on both sides of the equation.

Ex 1: Linear Equation with 1 solution

7y + 13 = 5y - 3
- 5y - 5y
2y + 13 = -3
-13 -13
2y = -16
2 2
y = -8

One Solution - intersecting lines

​Ex 2: Linear Equation with no solution

8 + 9p = 9p - 7
- 9p - 9p
8 = -7

This is NOT a true statement, therefore, there is NO Solution and the two lines are parallel.

​Ex 3: Linear Equation with infinitely many solutions

3(7r - 2) = 21r - 6
21r - 6 = 21r - 6
-21r -21r
-6 = -6
+6 +6
0 = 0

This IS a true statement, therefore, there are infinitely Many Solutions and the lines coincide ( or is the same line)

Fill in the Blank

Solve this linear equation in your notes and identify if it has 1 solution, no solution, or infinitely many solutions . Then determine if the lines intersect, are parallel, or coincide.

6n + 1 = 2n - 7

=

-

&

Open Ended

Solve this linear equation in your notes and identify if it has 1 solution, no solution, or infinitely many solutions . Then determine if the lines intersect, are parallel, or coincide.

5t + 7 = 5t - 8

Multiple Select

Solve this linear equation in your notes and identify if it has 1 solution, no solution, or infinitely many solutions . Then determine if the lines intersect, are parallel, or coincide.

2(2x - 2) = 4(x - 1)

• NO solution & parallel

• x = 2, One Solution &

  intersect

• -4 = -4

  Infinitely Many Solutions & coincide

• 4x = 4x

  Infinitely Many Solutions & coincide

Fill in the Blank

Identify if the following equation will have 1 solution, no solution or infinitely Many solution and if the lines intersect, are parallel, or coincide:

6w + 3 - 10w = 7w - 8

Multiple Choice

Solve the following equation for the unknown variable:

5t + 7 = 2t - 9

• No solution

• t = -8

• t = -4

• t = 5

Fill in the Blank

Solve this linear equation in your notes and identify if it has 1 solution, no solution, or infinitely many solutions . Then determine if the lines intersect, are parallel, or coinside.

2(3x + 6) = 3(2x - 6)

Multiple Choice

Solve this linear equation in your notes and identify if it has 1 solution, no solution, or infinitely many solutions . Then determine if the lines intersect, are parallel, or coinside.

8(3g + 2) - 3g = 3(5g - 4) - 2

• g = -5 & intersect

• g = -11 & coincide

• g = -6 & parallel

• g = 6 & coincide

Open Ended

When solving equations with variables on both sides, what is the ultimate goal for solving the equation?