Solve the system of equations using substitution.

First, isolate one of the variables. Since we are solving for , we are going to isolate  in the second equation. 

Replace with in our first equation.

Now we can solve to isolate .

After 20 minutes the first train would have traveled 20 miles. Let x be the amount of time elapsed. When 20 + 60x = 100x you will have the time in hours. 20 = 40x, x = 0.5 hrs. 0.5 hrs = 30 minutes. 

Solve by method of elimination. Multiply the first equation by 8 to eliminate the variable, m. Our first equation will then become .

By adding this new equation

with our second equation

We will see that our m cancel out. We can now solve for n. 

Now we have to plug in this value of n into any of our equations to find the value of m. 

Let's use the second equation.

Substitution needs to be used in order to solve this system of equations. From the second equation we know that ,

Substitute that into the first equation and solve.

You get

From there solve for y using the second equation.

Multiply the bottom equation by

Add the two equations together. This will allow you to cancel out  from an equation. From here, you can proceed to solve for .

From here, plug in the  value into the equation of your choice to solve for .

We are essentially presented with a system of equations. To solve for y, we will need to solve the system. The easiest way to solve this particular system is by adding the equations together.

First, multiply the second equation by 2.

Adding the two equations together will allow you to cancel the x values and solve for y.

If y equals 2, then 4y will be equal to 8.

To solve this problem, we translate the given information into two equations and then solve both simultaneously. If we let F represent the number of tanks that hold four fish and S represent the number of tanks that hold six fish, the problem tells us that F+S=15. The problem also tells us that 4F (the total number of fish in the 4-fish tanks) plus 6S (the total number of fish in the six-fish tanks) equals 70 (the total number of fish in the aquarium).

Thus we have the following system of equations:

F+S=15

4F+6S=70

Multiplying the first equation by -4 and combing it with the second gives 2S = 10, as seen below:

[-4F-4S=-60 (the first equation times -4)]

2S = 10

Therefore, S, the number of tanks that hold 6 fish, is 5.

Let  be the distance from home to church

Let  be the distance from church to mosque

To solve for y, first eliminate x by adding the two equations together such that the x’s factor out:

4x + 9y + 7 = 0

(–2)2x – (–2) 3y + (–2) 6 = (–2)0 (Multiply this equation by a factor of –2 so that 4x – 4x = 0)

Therefore, the two equations added together are:

(4x + 9y + 7 = 0) + (–4x –(–6)y +(–12) = 0) = (0 + 15y – 5 = 0)

y = 1/3

Let  be the number of dimes Julie has and  be the numbers of quarters she has. The number of dimes and the number of quarters add up to  coins. The value of all quarters and dimes is . We can then write the following system of equations:

To use substitution to solve the problem, begin by rearranging the first equation so that  is by itself on one side of the equals sign:

Then, we can replace  in the second equation with :

Distribute the :

Subtract  from each side of the equation:

Divide each side of the equation by :

Now, we can insert our value for  into the first equation and solve for :

Julie has  quarters and  dimes.