Can someone explain how these equations are formed?

There%26#039;s this word problem, that says:

An inheritance of $16,000 was divided among three investments yielding a total of $990 in interest per year. The interest rates were 5%, 6%, and 7%. Find the amount in each investment if 5% and 6% investments were $3000 and $2000 less than the 7% investment.

Then, I asked the teacher how you would solve this problem using the variables and linear systems. Then, she said that the equations are:

x + y +z = 16,000

0.05x + 0.06y + 0.07z = 990

x + 3000 = z

y + 2000 = z

I understand how she came up with the first and the second equation. However, I don%26#039;t understand the third and the fourth equations, and the last part of the problem, where it says %26quot; if 5% and 6% investments were $3000 and $2000 less than the 7% investment.%26quot;

Can someone explain to me how these equations are formed?

Can someone explain how these equations are formed?

I%26#039;ll try and rephrase the last part of the question:

%26quot;The amount invested in the 5% account was $3000 less than the amount invested in the 7% account, and the amount invested in the 6% account was $2000 less than the amount invested in the 7% account.%26quot;

The shortened this by ordered implication, since they put %26quot;5% and 6%%26quot; in that order, it%26#039;s assumed that, given the phrase %26quot;$3000 and $2000%26quot; after, the $3000 refers to the 5% and the $2000 refers to the 6%.

Sound good?

Can someone explain how these equations are formed?

well i think u hav 2 solve 4 x and y 1st 2 get z.

Can someone explain how these equations are formed?

Hi. It says the 5% investment was $3000 less than the 7% investment and that the 6% investment was $2000 less than the 7% investment. Since x represents the amount in the 5% investment and y represents the amount in the 6% investment, and z represents the amount in the 7% investment,

x = z - 3000 and y = z - 2000.

so, x + 3000 = z and y + 2000 = z.

Can someone explain how these equations are formed?

By saying that the 5% investment, x, is $3000 less than the 7% investment, z, it means:

x = z-3000

That%26#039;s the translation of the statement. Then if you add 3000 to both sides of the equation you%26#039;ll get what she wrote.

Another way to say it is

The 7% investment, z, is $3000 more than the 5% investment, z:

z = x + 3000

See how those are different ways of saying the same thing and their equivalent algebraic equations?

Can someone explain how these equations are formed?

x, y, and z are the amounts of money in each of the three equations. x is amount in 5%, y is in 6%, etc.

There are $3000 more in 7% than in 5%, and $2000 more in the 7% than in the 6%. That%26#039;s where the equations come from.

Can someone explain how these equations are formed?

The last two equations are from the last part of the problem, which is worded kind of stupidly.

It says that the amount that you invested at 5% is $3000 less than the amount you invested at 7%. And similarly, the amount you invested at 6% is $2000 less than the amount that you invested at 7%.

The second equation says that x is the amount at 5%, y is the amount at 6%, and z is the amount at 7%. Thus, you get the equations:

x = z - 3000 and y = z - 2000.

Rearrange these two equations and you get the last two equations that you didn%26#039;t understand.

Can someone explain how these equations are formed?

That last sentence is a bit confusing, and should divide up into 2 smaller sentences.

The 5% investment is $3000 less than the 7% investment.

The 6% investment is $2000 less than the 7% investment.

Your teacher calls the 5% investment %26quot;x%26quot;

the 6% is called %26quot;y%26quot;

and the 7% is called %26quot;z%26quot;

The third equation could also be x = z - 3000, which follows the sentence better, and is the same as her version, she just does an extra step.

The fourth equation could also be y = z - 2000, same reason.

If that%26#039;s not enough for you, then you probably need a lot more help from a tutor.