Unit 4, Lesson 8
Solving Any Linear Equation
Accelerated 7
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Solve each equation mentally.
Here are 4 problems. Select 2 to solve with your partner by taking turns describing a move, then writing an equivalent equation. For the other 2 problems, you and your partner should each solve 1 of the problems on your own, and then trade to check your answers.
Tyler says he invented a number puzzle. He asks Clare to pick a number, and then asks her to:
Clare says she now has -3. Tyler says her original number must have been a 3. How did Tyler know that?
Follow the same instructions starting with instead of a number. Explain or show your reasoning for why the last expression means that the person started with a number 6 greater than they ended with.
When we have an equation in one variable, there are many different ways to solve it. We generally want to make moves that get us closer to an equation that clearly shows the value that makes the equation true.
For example, or show that 5 and are solutions. Because there are many ways to do this, it helps to choose moves that leave fewer terms or factors.
If we have an equation like , adding -5 to each side will leave us with fewer terms. The equation then becomes .
Dividing each side of this equation by 3 results in the equivalent equation , which is the solution.
Or, if we have an equation like , dividing each side by 4 will leave us with fewer factors on the left. The equation then becomes .
Here is a list of valid moves that can help create equivalent equations that move toward a solution:
For example, suppose we want to solve .
From lots of experience, we learn when to use different valid moves that help solve an equation.