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Today, let’s look at writing equations and then solving them mentally.
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But before we start trying to write equations, we probably should ask the question, what is an equation?
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An equation is a sentence that contains two expressions separated by an equal sign.
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Here’s an example of that.
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Seven equals eight minus one.
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Here’s another example.
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18 equals 13 plus 𝑥.
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This equation contains a variable which is the 𝑥.
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This equation, 18 equals 13 plus 𝑥, is neither true or false until the variable is replaced with a number.
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Let’s say we replaced 𝑥 with a three.
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The equation is now false because 18 is not equal to 16.
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But when we’ve replaced the 𝑥 with the five, it becomes a true statement.
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We say that the equation here is true when 𝑥 equals five.
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We also say that five is the solution of the equation.
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A solution of an equation is the numerical value for the variable that makes the sentence true.
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In this case, five makes our sentence true.
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Solving an equation is the process of finding a solution.
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When we try to figure out what the variable is equal to, we call that solving the equation.
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Now, let’s take a look at how we solve an equation with mental math.
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So here’s an equation.
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𝑝 minus five equals 20.
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Remember that we wanna solve these problems mentally without writing things down.
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So we think some number take away five is equal to 20.
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And that number must be five more than 20.
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Therefore, 𝑝 must equal 25.
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𝑝 equals 25 is the correct answer.
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Here’s another example.
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If 𝑘 equals 60, is 28 plus 26 equal 𝑘 true or false?
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And again we wanna use a mental math strategy to solve this question.
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So you think about 28 and 26.
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They have an eight and a six in their ones position.
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And 60 has a zero in its one position.
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Eight plus six equals 14.
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And that means the solution to 𝑘 is going to have to have a four in the ones position.
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At this point, we don’t know what 𝑘 is.
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But we know that if 𝑘 equals 60, then this sentence must be false.
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We can say for sure that 28 plus 26 does not equal 60, which makes this statement false.
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Now, we are going to look at a question that asks us to write an equation.
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The total cost of a skateboard and knee pads is 247 dollars.
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If the skateboard costs 235 dollars, write an equation to find the cost of the kneepads, 𝑘.
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Then solve the equation.
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The problem is asking us to do two things, write an equation and then solve the equation that we write.
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We’re given three different pieces of information.
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The total cost is 247 dollars.
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The skateboard costs 235 dollars.
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And the missing cost, the cost that we don’t know, is 𝑘, and that’s the cost of the kneepads.
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We know that the total cost equals the skateboard price plus the kneepads price.
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Let’s substitute these two pieces with the information that we already know.
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Total cost is 247.
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Skateboard costs 235.
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What should we put here for the knee pads price?
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Because we don’t know the knee pads price, we need to substitute a variable here.
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In our case, that’s where the 𝑘 goes.
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Our equation becomes 247 equals 235 plus 𝑘.
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Step one of the problem is complete.
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This is an equation we can use to solve step two.
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And this part can be done mentally.
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What added to 235 would equal 247?
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Well, if you add ten, you get from 235 to 245 add two more to that and now you’re at 247.
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Put both of those numbers you added together, 10 and two, and you realise that 𝑘 must be equal to 12.
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𝑘 is equal to twelve.
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And that means the kneepads cost 12 dollars.
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Our final answer here is 𝑘 equals 12 dollars.