Need help with this type of problem.

[Colto]

I am not sure what this type of problem is called, or how to solve it. Any help, even just what the type of problem is called would be much appreciated.
"Solve this problem by defining a variable and solving an equation. Write your solution in a sentence.

West High School's population is 250 students fewer than twice the population of East High School. The two schools have a total of 2858 students. How many students attend East High School?"

 

[Otis]

I am not sure what this type of problem is called, or how to solve it.

Hi Colto. It's a beginning algebra problem, and there are different methods to solve it. One way is to write a system of two equations and then solve the system using the substitution method.

Have you solved linear equations before? For example, do you know how to solve an equation like this?

276 - x = 2x - 45

Are you familiar with translating English statements into mathematical expressions? For example, can you write an algebraic equation that says: "X is 250 fewer than twice Y"?

Once you provide some context about your current knowledge, we'll know what kind of help you need. Thanks!

 

[Colto]

Hi Colto. It's a beginning algebra problem, and there are different methods to solve it. One way is to write a system of two equations and then solve the system using the substitution method.

Have you solved linear equations before? For example, do you know how to solve an equation like this?

276 - x = 2x - 45

Are you familiar with translating English statements into mathematical expressions? For example, can you write an algebraic expression that says: "X is 250 fewer than twice Y"?

Once you provide some context about your current knowledge, we'll know what kind of help you need. Thanks!

Thank you, no I am not sure how to solve "276 - x = 2x - 45". I am familiar with "X is 250 fewer than twice Y" however, x=2y-250

 

[Otis]

I am not sure how to solve "276 - x = 2x - 45".

I understand, but it means you're not ready to solve problems like this. Would jogging your memory help? Let's see whether you can understand and follow these three steps, for solving the equation.

1. Add x to each side and simplify
2. Add 45 to each side and simplify
3. Divide each side by 3 and simplify

If not, please tell me what you're unsure about. Otherwise, show what you can try.

 

[Colto]

I understand, but it means you're not ready to solve problems like this. Would jogging your memory help? Let's see whether you can understand and follow these three steps, for solving the equation.

1. Add x to each side and simplify
2. Add 45 to each side and simplify
3. Divide each side by 3 and simplify

If not, please tell me what you're unsure about. Otherwise, show me what you can try.

Oh yes, that did help. 276 - x = 2x - 45 | 276 = 3x - 45 | (276+45)= 3x | 321 = 3x | 107 = x Thank you, I now understand that, however I am still unsure what steps to take to solve x = 2y - 250.

 

[Otis]

Very good. (We won't be solving that other equation, but we'll be writing something similar to solve your exercise.)

The first step with word problems in algebra is to define variables. They say to use one variable, but I'm going to use two at first -- because there are two unknown quantities -- and I'll explain how to use one variable later. We could pick x and y, but I'm going to pick symbols that help me remember what they stand for.

Let E = student population at East High School

Let W = student population at West High School

Now that we have symbols to represent those student populations, we can translate the given relationships from English into algebraic equations.

Use symbols E and W to write an equation representing "West High School's population is 250 students fewer than twice the population of East High School".

Do the same with "The two schools have a total of 2858 students".

Please show me your two equations.

 

[Colto]

W = 2e - 250 | W + E = 2858

 

[Otis]

W = 2e - 250 | W + E = 2858

Good job, but let's not treat symbols e and E the same. The equations above comprise your "system of two equations".

W = 2E - 250

That equation shows we have two ways to represent the number of students at West High School. In other words, the symbol W may be replaced with the expression 2E-250 anywhere W appears because both represent the same number.

The substitution method for solving the system of two equations involves substituting the expression 2E-250 for variable W in the second equation:

W + E = 2858

Once you do that, you'll obtain a new equation containing only the variable E. That will be the equation to solve for E (using steps similar to the first example you solved earlier).

Give it a go, and show us what you get.

 

[Colto]

2E - 250 + E = 2858 | 3E - 250 = 2858

 

[Otis]

Great. You've partially solved, by combining the two E-terms to get 3E.

3E - 250 = 2858

We want the E-term on one side, and we want the constants combined on the other side.

Think: What operation can we do to both sides to get 3E all by itself on the left?

If you're not sure, then go back and look at what you did earlier to the equation 276=3x-45 to get 3x all by itself.

 

[Colto]

Ohh, add 250! 3E - 250 = 2858 | 3E = 3108 | 3108/3 = 1,036 | E = 1,036 | Thank you so much, you are really a great person to take your free time to help people with math for no cost! I really appriciate everything you have done!

 

[Otis]

You're welcome. Let's check the answer, and then I'll finish with some commentary.

E=1036 is the proposed solution. So that would mean:

W + 1036 = 2858

W = 2858 - 1036 = 1822

To be solutions, the numbers E=1036 and W=1822 must satisfy both equations in the system:

W = 2E - 250

1822 = 2(1036) - 250

1822 = 2072 - 250

1822 = 1822

It checks, and the answer is "1,036 students attend East High School".

---

Here are comments about using one variable versus two.

We know at the beginning that there are 2858 total students. Subtracting E from the total gives W, and subtracting W from the total gives E, yes?

2858 - E = W

Were we to consider that fact mentally at the beginning, we could have assigned expressions for the student populations like this, instead:

Let E = student population at East High School

Then 2858-E = student population at West High School

That way, we don't need to write the second equation (E+W=2858). We don't need W at all. We translate only the first English statement:

2858 - E = 2E - 250

and solve directly for E without ever having written W.

It's a mental shortcut, but I wanted you to see the substitution method. Also, I think it's better for beginning students to work with two symbols, until they have enough experience to realize the mental shortcut on their own. Many problems would ask for both numbers, anyway, yet even when they don't it's still nice to have both numbers for checking purposes.

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[Otis]

If you'd like to practice, here's another exercise that can be answered similarly (using two symbols).

I have $1.25 worth of dimes and quarters in my pocket. There are a total of eight coins. How many dimes do I have?

Cheers!

 

[Otis]

This YouTube video shows a number of worked examples using the substitution method.

It's one of many videos available here. Search that Algebra1 index page for the word "system", and you'll see many video links for additional examples, as well as other methods for solving systems of equations.

Have fun!

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