Warm Up HW check

Each cylinder is equivalent to two blocks. The balance problem is equivalent to x + 5 = 3x + 1, where a cylinder represents x and the cubes represent the whole numbers. This is an example of an equation with variables on both sides, the type you will solve today.

Exploration #1- work with your seat partner If you finish before time is up…keep going!

1.3 Solving Equations with Variables on Both Sides Today’s Learning Goal is: Students will be able to solve linear equations, identify special solutions and use linear equations to solve real-life problems.

To solve an equation with variables on both sides, simplify one or both sides of the equation, then use inverse operations to collect the variable terms on one side, constant terms on the other and then isolate the variable.

Examples 1.10 – 4x = -9x 10 -4x + 4x = -9x + 4x Add 4x to both sides 10 = -5x Combine Like Terms 10=-5x Divide both sides by = x

Examples 2. 3(3x – 4) = ¼ (32x + 56) 9x – 12 = 8x + 14 Distribute the 3 and the 1/4 9x = 8x Add 12 to both sides 9x = 8x +26 Combine Like Terms 9x – 8x = 8x – 8x + 26 Subtract 8x from both sides x = 26 Combine Like Terms

You try 1.-2x = 3x ½ (6h – 4) = -5h ¾ (8n + 12) = 3 (n – 3) x = -2 h = 3/8 n = 0

Special Solutions of Linear Equations Equations do not always have one solution. An equation that is true for all values of the variable is an identity and has infinitely many solutions. An equation that is not true for any value of the variable has no solution.

Examples This is never true so the equation has no solution

Examples Identify the Number of Solutions (4y + 1) = -8y – 2 -8y - 2 = -8y – 2 +8y +8y -2 = -2 This is always true, so the equation is an identity and has infinitely many solutions.

You Try! 4.4(1 – p) = -4p m – m = 5/6 (6m – 10) 6.10k +7 = -3 – 10k 7.3 (2a – 2) = 2 (3a – 3) Infinite solutions No solution Infinite solutions

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Examples Modeling with Mathematics 4. A boat leaves New Orleans and travels upstream on the Mississippi River for 4 hours. The return trip takes only 2.8 hours because the boat travels 3 miles per hour faster downstream due to the current. How far does the boat travel upstream?

Examples -Understand the Problem You are given the amounts of time the boat travels and the difference in speeds for each direction. You are asked to find the distance the boat travels. -Make a Plan Use the Distance Formula (distance = rate * time) to write expressions that represent the problem.

Examples -Solve the Problem distance upstream = rate * time distance downstream = rate * time rate * time = rate * time Is the answer reasonable?

You Try! 17.5 mi

Workbook p. 15 & 16 with your seat partner *each student does and shows the work in their own workbook – your partners are there to help you!