Solving Equations - Math

1st equation:

2nd equation:

Subtract the 2nd equation from the 1st equation to eliminate the "2y" from both equations and get an answer for x:

Plug the value of into either equation and solve for :

The break-even point occurs when the .

The equation to solve becomes

so the break-even point is .

So the equation to solve becomes , or must be sold to make a profit of .

The two equations in this system can be combined by addition or subtraction to solve for and . Isolate the variable to solve for it by multiplying the top equation by so that when the equations are combined the term disappears.

Divide both sides by to find as the value for .

Substituting for in both of the two equations in the system and solving for gives a value of for .

The first step in solving this equation is to distribute the 2 through the parentheses. This gives us:

Next, we subtract 6 from both sides, in order to get the variable alone on one side of the equation:

Finally, we divide both sides by 2 to solve for :

The first step in solving this equation is to combine the like terms. That is, move all of the terms with an in it to one side of the equation, and all the terms without an to the other side. Let's begin with moving the terms to the left side of the equation. We do this by subtracting from each side:

Next, we combine the terms without an on the right side. We do this by subtracting 3 from both sides:

Finally, we divide both sides of the equation by 2 to solve for :

The first step to solve this equation is expand out the right-hand side of the equation by multiplying the 4 through the parentheses:

Next, we combine like terms. That is, we put all of the terms with an on one side of the equation and all the terms without on the other. We'll start by subtracting from both sides:

And now we'll move the non- terms to the right-hand side by subtracting 2 and 10 from both sides:

Combining these terms, this simplifies to:

Next, we divide both sides by 8:

Which gives:

The first step in solving this equation is to distribute the 3 and the 4 through the parentheses:

Simplify:

Now, we want to get like terms on the same sides of the equation. That is, all of the terms with an should be on one side, and those without an should be on the other. To do this, we first subtract from both sides:

Simplify:

Now, we subtract 6 from both sides:

The first step in solving this equation it to subtract 3 from both sides, so that the term with is alone on one side of the equals sign.

which simplifies to:

From here, we multiply both sides by 2:

This gives:

The first step to solving this equation is to combine the terms with in them on the left-hand side of the equation. This gives:

Next, we can subtract from both sides of the equation:

Which simplifies to:

Then, we subtract 2 from both sides of the equation:

And to finish, we now divide both sides by 2:

Which simplifies to:

The first step in solving this equation is to distribute the 3 through the parentheses on the left-hand side of the equation:

Which gives:

Now, in order to get rid of the fraction, we can multiply the whole equation by 3:

We can now combine the like terms on the left-hand side, which gives:

Now, we subtract from both sides:

Next, we subtract 18 from both sides:

Last, we divide both sides by 7 to solve for :

Which gives,

Simply plugging in -2 into each answer choice will determine the correct answer:

The key to solving this problem is to remember the order of operations and that negative numbers squared are positive, while negative numbers raised to the third power are negative.

When solving the equation , observe that . Taking the cube root of 216 gives 6. Thus,

In the box, there are kilograms of baseballs and kilograms of footballs. In total, there are kilograms of balls. The total weight of the box is 48 kilograms, so Since there are kilograms of footballs, the total weight of the footballs in the box is equal to:

First, notice that we can factor into the form (a-b)(a+b). We are told that a-b=3, so we can substitute that into the first equation.

If we divide both sides by 3, we can obtain the value of a+b.

We now have a system of equations: a-b = 3, a+b = 11. We will solve this system by elimination. If we add the two equations together, we obtain the following:

.

Divide both sides by 2.

Going back to the equation a-b = 7, we can solve for b.

Add b to both sides.

Subtract 3 from both sides.

Ultimately, the question asks us to determine the value of .

= .

The answer is 65.

Equation 1:

Equation 2:

Solve equation 2 for X:

Substitute into Equation 1:

Solve for y: ,

Take the answer for y and plug it back into either original equation to find x:

Equation 1:

Equation 2:

Solve Equation 2 for y:

Substitute into Equation 1:

Plug x back into either original equations and solve for y:

Since we have variables on both sides, we first subtract from both sides.

This leaves us with .

Subtracting 2 from both sides, we get .

First, use the distributive property to multiply out the left side:

Then, add on both sides:

Subtract 2 from both sides:

Divide both sides by 10 to get , or .