The larger of two numbers is 5 more than twice the smaller. If
the smaller is subtracted from the larger, the result is 12. Find
the numbers.

Make sure that you read the question carefully several times.

Since we are looking for two numbers, we will let

*x* = the smaller number

*y* = the larger number

**Equation (1):**

**Equation (2):**

This is a system of linear equations with two variables, which can
be found in **Tutorial 19: Solving
Systems of Linear Equations in Two Variables.**

**No simplification is needed for this problem. **

**I'm going to go ahead and have us use the substitution
method as shown in Tutorial 19: Solving Systems of Linear Equations in
Two Variables to solve this.**

**Solve one equation for either variable**

Since equation (1) is nicely solved for y, we will use that to substitute
into the other equation.

*y *= 5 + 2*x *(1)

**Substitute what you get above into the other
equation AND solve for the remaining variable.**

**Substituting in 5 + 2 x for y into equation (2) and solving for x we get:**

***Inverse of add 5 is sub. 5**

**Using equation (1) to plug in 7 for x and
solving for y we get:**

You will find that if you plug the ordered pair (7, 19) into BOTH equations
of the original system, this is a solution to BOTH of them.

**Final Answer:**

**The numbers are 7 and 19.**

It takes a boat 2 hours to travel 24 miles downstream and 3 hours to
travel 18 miles upstream. What is the speed of the boat in still
water and of the current of the river?

Make sure that you read the question carefully several times.

Since we are looking for two different rates, we will let

*x* = rate of the boat

*y* = the rate of the current

Since this is a rate/distance problem, it might be good to organize our information using the distance formula.

Keep in mind that the rate of the current is affecting the overall speed.

**When the boat is going upstream, it will be going slower. That
rate will be x - y. **

**When the boat is going downstream, it will be going faster.
That rate will be x + y.**

**Equation (1):**

**Equation (2):**

This is a system of linear equations with two variables, which can
be found in **Tutorial 19: Solving
Systems of Linear Equations in Two Variables.**

**Simplify if needed. **

**We can simplify this by dividing both sides of equation (1) by 3
and equation (2) by 2:**

***Div. both sides of eq. (2) by 2**

At this point, you can use any method that you
want to solve this system. I'm going to use the **elimination
method as discussed in Tutorial 19: Solving Systems of Linear Equations
in Two Variables.**

**Multiply one or both equations by a number
that will create opposite coefficients for either x or y if needed AND
add the equations.**

**Since we already have opposite coefficients
on y, we can go right into adding equations
(3) and (4) together:**

*** y's dropped out**

**Solving for x we get:**

**Using equation (4) to plug in 9 for x and
solving for y we get:**

***Inverse of mult. by -1 is div. by -1**

You will find that if you plug the ordered pair (9, 3) into BOTH equations
of the original system, this is a solution to BOTH of them.

**Final Answer:**

**The boat speed is 9 mph and the rate of the current is 3 mph.**

A student has money in three accounts that pay 5%, 7%, and 8%, in annual
interest. She has three times as much invested at 8% as she does
at 5%. If the total amount she has invested is $1600 and her interest
for the year comes to $115, how much money does she have in each account?

Make sure that you read the question carefully several times.

Since we are looking for three different amounts, we will let

*x* = the**amount invested at 5%**

*y *= the**amount invested at 7%**

*z *= the**amount invested at 8%**

**Equation (1):**

**Equation (2):**

**Equation (3):**

This is a system of linear equations with three variables, which can
be found in **Tutorial 20:
Solving Systems of Linear Equations in Three Variables.**

**Simplify and put all three equations in the
form Ax + By + Cz = D if needed.**

**Equation (1) needs to be put in the correct form and equation (3)
needs to be multiplied by 100 to get rid of the decimals:**

***Mult. both sides of eq. (3)****by 100**

Since y is already eliminated in equation (4), it would be quickest and easiest to eliminate y.

**We can use equation (4) as one equation with y eliminated**

**Multiplying equation (2) by -7 and adding it to equation (5) we get:**

*** y's have opposite
coefficients**

*** y's dropped out**

**Putting those two equations together we get:**

**Multiplying equation (6) by -1 and then adding that to equation (4)
we get:**

*** z's have opposite
coefficients**

*** z's dropped out**

***Inverse of sub. 900 is add 900**

**Using equation (2) to plug in 300 for x and 900 for z we get:**

***Inverse of add 1200 is sub. 1200**

You will find that if you plug the ordered triple (300, 400, 900) into
ALL THREE equations of the original system, this is a solution to ALL THREE
of them.

**Final Answer:**

**$300 is invested at 5%, $400 is invested at 7% and $900 is invested
at 8%.**

Last revised on July 10, 2011 by Kim Seward.

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