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Alright, we're dealing with a few values here, so let's give them some labels to save us some trouble down the road. We'll call the number of messages sent by Maria <em>m</em>, the number sent by Bill <em>b</em> and the number sent by Change (is that a real name?) <em>c</em>. We don't know exactly what each number is, but let's take a look at what information they do give us.
Change sent 2 times as many messages as Bill, or, using our variable for Change and Bill:
We're also given that Maria sent 7 messages more than Bill, which we can represent with:
Notice that <em>m</em> and <em>c</em> are both in terms of <em>b</em>. We can use this for our next step. We're given at the beginning that together, Maria, Bill, and Change sent 71 messages over the weekend. As an equation using all of our variables, this translates to:
Since <em>m </em>and <em>c </em>are both in terms of <em>b</em>, we can substitute those expressions in and solve for <em>b</em>:
Now that know that Bill sent 16 texts, we can find the numbers for Change and Maria:
So, Bill sent 16 texts, Maria sent 23, and Change sent 32.
Change sent 2 times as many messages as Bill, or, using our variable for Change and Bill:
We're also given that Maria sent 7 messages more than Bill, which we can represent with:
Notice that <em>m</em> and <em>c</em> are both in terms of <em>b</em>. We can use this for our next step. We're given at the beginning that together, Maria, Bill, and Change sent 71 messages over the weekend. As an equation using all of our variables, this translates to:
Since <em>m </em>and <em>c </em>are both in terms of <em>b</em>, we can substitute those expressions in and solve for <em>b</em>:
Now that know that Bill sent 16 texts, we can find the numbers for Change and Maria:
So, Bill sent 16 texts, Maria sent 23, and Change sent 32.
Notice that based on the diagram, all of the sides of the triangle are the same length. (The dashes on the sides of the triangle show that the sides are of congruence.) This means that the triangle is an equilateral triangle. An important thing to note about equilateral triangles is that they all have angles of 60°. All angles in equivalent triangles are 60°.
This means that we can set equal to 60 to solve for
:
- Set up
- Add 3 to both sides
- Divide both sides by 7
Our answer is x = 9.