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Solving Equations with Variables on Both Sides of the Equal Sign This lesson shows you step by step how to solve equations with variables on both sides of the equal sign. Just sit back and enjoy the equations being solved. After the show, you will have the option of going more slowly through the four examples.

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Solving equations with variables on both sides of the equal sign. 7x – 2 = 2x + 8 2x2x2x2x -- 7x - 2x – 2 =2x - 2x + 8 2x is smaller than 7x so it is the smaller variable term. Subtract 2x from both sides of the equation Combine like terms 5x – 2 = 8 This is just a two-step equation. Now solve for x Find the smaller variable term

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Follow the two-step procedure to solve for x 5x – 2 = 8 5x – 2 + 2 = 8 + 2 5x = 10 55 x = 2 Isolate the variable by adding 2 to both sides of the equation. Then divide both sides by 5. Solve for x. What does this solution mean? 2-12-1 1-11-1

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What the solution means… When you substitute x=2 in the equation, what happens? y = 7x - 2 y = 2x + 8 GEOMETRICALLY 7x – 2 = 2x + 8 Where do they cross? x = 2 7x – 2 = 2x + 8 7(2) - 2 = 2(2) + 8 14 - 2 = 4 + 8 12 = 12 ALGEBRAICALLY You make the equation TRUE.

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That wasn’t so bad Mr. Anderson, let’s try another!

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A second equation with variables on both sides of the equal sign. 5 - 3x = 2x + 10 3x3x3x3x ++ 5 - 3x + 3x =2x + 3x + 10 -3x is smaller than 2x. It’s Negative! Subtract -3x from both sides of the equation. That’s the same as adding 3x. Combine like terms 5 = 5x + 10 This is just a two-step equation. Now solve for x Find the smaller variable term

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Follow the two-step procedure to solve for x 5 = 5x + 10 5 - 10 = 5x + 10 - 10 -5 = 5x 55 -1 = x Isolate the variable by subtracting 10 from both sides of the equation. Then divide both sides by 5. Solve for X What does this solution mean? 1-11-1 - 1

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What the solution means… When you substitute x=-1 in the equation, what happens? y = 5 – 3x y = 2x + 10 GEOMETRICALLY 5 – 3x = 2x + 10 Where do they cross? x = -1 5 – 3x = 2x + 10 5 – 3(-1) = 2(-1) + 10 5 + 3 = -2 + 10 8 = 8 ALGEBRAICALLY You make the equation TRUE.

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WOW, I think I can do that! But, Mr. Anderson, is that all there is to it? Good question. Let’s watch and learn!

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A special equation with variables on both sides of the equal sign. 3x – 4 = 1 + 3x 3x3x3x3x -- 3x - 3x – 4 =1 + 3x – 3x 3x is the smaller variable term. So, subtract 3x from both sides of the equation Combine like terms -4 = 1 WHOA! Where did the x go? Find the smaller variable term What is the answer?

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-4 = 1 Think about what this equation says. “Negative four is equal to one.” BUT, negative four is NOT equal to one. Therefore, the answer is… NO SOLUTION What does THIS solution mean?

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What the solution means… For the given equation y = 3x - 4 y = 1 + 3x GEOMETRICALLY 3x – 4 = 1 + 3x Where do they cross? 3x – 4 = 1 + 3x There exist NO value for x which makes the left side of equation equal the right side of the equation. ALGEBRAICALLY You can’t make the equation TRUE. They will NEVER cross. They are parallel.

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So far, we have seen two types of solution. 1.One solution, like x = 2. 2. No solution, lines are parallel. Let’s watch one more type of solution.

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A third type of equation with variables on both sides of the equal sign. 2x + 5 = 5 + 2x 2x2x2x2x -- 2x - 2x + 5 =5 + 2x - 2x 2x is the smaller variable term. Subtract 2x from both sides of the equation Combine like terms 5 = 5 Again, the x – term has disappeared. Find the smaller variable term

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5 = 5 Think about what this equation says. “Five is equal to five.” Duh, five is equal to five. Great. Therefore, the answer is… IDENTITY What does THIS solution mean?

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What the solution means… When you substitute ANY VALUE in the equation, what happens? y = 2x + 5 y = 5 + 2x GEOMETRICALLY 2x + 5 = 5 + 2x Where do they cross? 2x + 5 = 5 + 2x Say, x = 4 2(4) + 5 = 5 + 2(4) 8 + 5 = 5 + 8 ALGEBRAICALLY You make the equation TRUE. 13 = 13 Say, x = -3 2(-3) + 5 = 5 + 2(-3) -6 + 5 = 5 - 6 -1 = -1 OR No matter what number you plug in, you get a true equation! They cross everywhere. They are IDENTICAL lines!

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What Have We Learned Today? When Solving Equations with Variables on Both Sides of the Equal Sign Find the smaller variable term and subtract. Combine like terms. Then, just solve the two-step equation, normally. There are THREE types of solutions: - One solution, like x = 2. The lines intersect once. - No solution. The lines are parallel. -Identity. The lines coincide. They are the same.

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