Solve the equation.

$\displaystyle -5(3x+4)=-200$ 

Step 1:

Distribute the $\displaystyle -5$ on the left side of the equation.

$\displaystyle -15x-20=-200$

Step 2:

Add $\displaystyle 20$ to both sides of the equation to isolate the term with the $\displaystyle x$ variable.

$\displaystyle -15x=-180$

Step 3:

Divide $\displaystyle -15$ by both sides of the equation to solve for $\displaystyle x$.

$\displaystyle x=12$

Solution: $\displaystyle x=12$

Solve the equation.

$\displaystyle 4+6x=4x-20$

 Step 1:

Subtract $\displaystyle 4$ from both sides of the equation to isolate the term with the $\displaystyle x$ variable.

 $\displaystyle 6x=4x-24$

Step 2: 

Subtract $\displaystyle 4x$ from both sides of the equation to have only one term with the $\displaystyle x$ variable on one side.

 $\displaystyle 2x=-24$

Step 3:

Divide $\displaystyle 2$ by both sides of the equation to solve for $\displaystyle x$.

 $\displaystyle x=-12$

Solution: $\displaystyle x=-12$

When the sum of five times a number and two is doubled, the result is $\displaystyle 74$. What is the number?

Part 1: Writing the equation

In order to find the value of the number, we must take this word problem and convert it into a concrete equation that we can solve. Let's analyze the problem in order to understand what is being asked. In the following problem, the key words will be bolded.

"when the sum of five times a number and two is doubled, the result is $\displaystyle 74.$"

Sum: 

This means that you will have to add. This is written down as the $\displaystyle +$ symbol.

Five times a number: 

This means that you will multiply a number by $\displaystyle 5$. Let's let this unknown number equal $\displaystyle x$. The reason that we are doing this is to make it easier for us to convert the word problem into an actual equation. It is much more simple to calculate "$\displaystyle 5x$" than it is to calculate "five times a number."

And two: 

This means that there is another part to this equation. You must perform an operation to $\displaystyle 5x$ and $\displaystyle 2$. As stated in the problem, you must find the sum. This can be written out as $\displaystyle 5x+2$. 

Is doubled:

This means that you must multiply everything you have written down so far by $\displaystyle 2$. The simplest way to write this in algebra is to put $\displaystyle 5x+2$ in parentheses, and have the $\displaystyle 2$ outside of the parentheses, as shown below:

$\displaystyle 2(5x+2)$

The result is $\displaystyle 74$:

This means $\displaystyle =74$.

We now have all of the information necessary to write a concrete equation from this word problem.

 $\displaystyle 2(5x+2)=74$

Part 2: Solving the equation

We can solve  $\displaystyle 2(5x+2)=74$ just as we would for any other equation.

Step 1:

Distribute the two throughout all of the terms inside the parentheses.

 $\displaystyle 10x+4=74$

Step 2:

Subtract $\displaystyle 4$ from both sides of the equation.

 $\displaystyle 10x=70$

Step 3:

Divide $\displaystyle 10$ by both sides of the equation to solve for $\displaystyle x$.

 $\displaystyle x=7$

Solution: $\displaystyle x=7$

Part 1: Writing the equation

The first step to solving this problem is writing down a solvable equation.

In this cell phone plan, you must pay a $100 down payment. In other words, no matter how long or how little you talk on the phone, you must pay this $100. Another part that is added to this $100 is fifty cents per hour of talking time. Because we are looking for the amount of time Corey can talk on the phone, let's let $\displaystyle x=$ the number of hours. Fifty cents can be represented by $\displaystyle 0.5$. This must be multiplied by $\displaystyle x$ to find out how long he can speak on the phone with this plan.

So far, we have $\displaystyle 100+0.5x$.

According to Corey's budget, he only has $300 to spend on this plan. Therefore, we can set this equal to $\displaystyle 300$, and solve for $\displaystyle x$.

$\displaystyle 100+0.5x=300$

Part 2: Solving the equation

$\displaystyle 100+0.5x=300$

Subtract 100 on both sides of the equation to isolate the term with the variable $\displaystyle x$.

$\displaystyle 0.5x=200$

Divide both sides of the equation by $\displaystyle 0.5$ to solve for $\displaystyle x$.

$\displaystyle x=400$

Solution: $\displaystyle 400$ hours.

$\displaystyle 2x+38=0$

In single variable algebraic equations you want to isolate the unknown variable, which is usually a letter, and in this case $\displaystyle x$, then solve for it. It is generally better to do simple operations first (e.g., addition and subtraction), so you aren't left with more complicated fractions.

1) The first step would be to move the 38 to the other side of the equation by subtracting 38 from each side (adding -38).

Giving you:

$\displaystyle 2x=-38$

2) Now to isolate $\displaystyle x$, you would divide each side of the equation by 2.

Giving you:

$\displaystyle x=-19$

3) In order to verify the result you can substitute your answer in for $\displaystyle x$ and simplify it through operations. If you get the expression 0=0 or any other true expression it is correct:

$\displaystyle \\2(-19)+38=0 \\-38+38=0 \\0=0$

Solve for x and y by using substitution. Solve the first equation for x.  

$\displaystyle x=\frac{72+4y}{6}$

Next substitute x into the second equation and simplify,

$\displaystyle 7(\frac{72+4y}{6})-y=62$

$\displaystyle 84+4.6666y-y=62$

$\displaystyle 3.6666y=-22$

$\displaystyle y=-6$

Now use the value for y to solve for x.

$\displaystyle x=\frac{72+4(-6)}{6}=8$

Solve for x and y by using substitution. Solve the first equation for x.  

$\displaystyle x=\frac{16-5y}{3}$

Next substitute x into the second equation and simplify,

$\displaystyle 7(\frac{16-5y}{3})-4y=6$

$\displaystyle 37.333-11.666y-4y=6$

$\displaystyle -15.666y=-31.333$

$\displaystyle y=2$

Now use the value for y to solve for x.

$\displaystyle x=\frac{16-5(2)}{3}=2$

Solve for x and y by using substitution. Solve the first equation for x.  

$\displaystyle x=\frac{49-3y}{4}$

Next substitute x into the second equation and simplify,

$\displaystyle 5.5(\frac{49-3y}{4})+32y=39.5$

$\displaystyle 67.375-4.125y+32y=39.5$

$\displaystyle 27.875y=-27.875$

$\displaystyle y=-1$

Now use the value for y to solve for x.

$\displaystyle x=\frac{49-3(-1)}{4}=13$

Solve for x and y by using substitution. Solve the first equation for x.  

$\displaystyle x=\frac{15+3y}{2}$

Next substitute x into the second equation and simplify,

$\displaystyle 8(\frac{15+3y}{2})+2y=4$

$\displaystyle 60+12y+2y=4$

$\displaystyle 14y=-56$

$\displaystyle y=-4$

Now use the value for y to solve for x.

$\displaystyle x=\frac{15+3(-4)}{2}=1.5$

Solve for x and y by using substitution. Solve the first equation for x.  

$\displaystyle x=15+y$

Next substitute x into the second equation and simplify,

$\displaystyle 2(15+y)+2y=14$

$\displaystyle 30+2y+2y=14$

$\displaystyle 4y=-16$

$\displaystyle y=-4$

Now use the value for y to solve for x.

$\displaystyle x=15+(-4)=11$