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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.**

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

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**Follow the two-step procedure to solve for x**

Isolate the variable by adding 2 to both sides of the equation. 5x = 10 1 - 2 - 1 5 5 Then divide both sides by 5. x = 2 Solve for x. What does this solution mean?

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**What the solution means…**

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

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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.**

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

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**Follow the two-step procedure to solve for x**

Isolate the variable by subtracting 10 from both sides of the equation. -5 = 5x -1 - 1 1 - 5 5 Then divide both sides by 5. -1 = x Solve for X What does this solution mean?

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**What the solution means…**

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

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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.**

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

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**-4 = 1 NO SOLUTION Think about what this equation says.**

“Negative four is equal to one.” BUT, negative four is NOT equal to one. NO SOLUTION Therefore, the answer is… What does THIS solution mean?

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**What the solution means…**

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

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**So far, we have seen two types of solution.**

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.**

Find the smaller variable term 2x + 5 = 5 + 2x 2x is the smaller variable term. 2x = 5 + 2x - 2x - 2x Subtract 2x from both sides of the equation 2x - 2x + 5 = 5 + 2x - 2x Combine like terms 5 = 5 Again, the x – term has disappeared.

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**5 = 5 IDENTITY Think about what this equation says.**

“Five is equal to five.” Duh, five is equal to five. Great. IDENTITY Therefore, the answer is… What does THIS solution mean?

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**What the solution means…**

ALGEBRAICALLY GEOMETRICALLY When you substitute ANY VALUE in the equation, what happens? 2x + 5 = 5 + 2x Say, x = 4 Say, x = -3 OR 2(4) + 5 = 5 + 2(4) 2(-3) + 5 = 5 + 2(-3) 8 + 5 = 5 + 8 = 5 - 6 13 = 13 -1 = -1 2x + 5 = 5 + 2x y = 2x + 5 y = 5 + 2x You make the equation TRUE. Where do they cross? 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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