Solving Equations

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Math › Solving Equations

Questions 1 - 10

An art show wants to focus on sculpture but will also accept paintings and drawings. The show has room for pieces and will display them in a ratio of scupltures to painting to drawings. How many sculptures will be displayed?

Explanation

Since the paintings and the drawings have the same ratio, they will have the same number on display.

Let number of paintings displayed = number of drawings displayed.

Then, number of sculptures displayed.

Set up a proportion:

$\frac{sculptures}{drawings}=\frac{140 - 2x}{x}=\frac{3}{2}$

Solve by cross multiplying to get or .

Then add to both sides to get .

Then divide by to get .

So there are drawings and paintings in the show, leaving the last spots for sculptures.

Solve the equation for .

No solution

Explanation

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

This leaves us with .

Subtracting 2 from both sides, we get .

Solve the system of equations.

None of the other answers are correct.

Explanation

Isolate in the first equation.

Plug into the second equation to solve for .

Plug into the first equation to solve for .

Now we have both the and values and can express them as a point: .

Solve the following equation for :

$x=\frac{1}{2}$

Explanation

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:

$\frac{2x}{2}=\frac{4}{2}$

Which simplifies to:

Solve:

Explanation

Add the two equations together to get . Then divide by to get .

Now substitute this value into one of the original equations to get .

Next subtract from both sides to get .

Now divide both sides by to get .

Therefore the answer is .

Solve the pair of equations for x and y:

Explanation

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:

Lilac paint is made by mixing one part blue paint and one part red pain with two parts white paint. How many gallons of white paint should be mixed with two quarts of blue paint?

Explanation

Set up a proportion where number of quarts of white paint.

$\frac{white}{blue} = \frac{x; qt}{2; qt}= \frac{2}{1}$ and solve by cross multiplying to get , and .

Solve the pair of equations for x and y:

Explanation

Equation 1:

Equation 2:

Solve Equation 2 for y:

Substitute into Equation 1:

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

Solve for

$y = \frac{10}{7}$

$y = \frac{1}{7}$

$y = \frac{7}{5}$

Explanation

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 .

Solve the following equation for :

Explanation

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

$3\cdot x+3\cdot2 = 4\cdot x-4\cdot 3$

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: