### All ACT Math Resources

## Example Questions

### Example Question #1 : How To Find Out When An Equation Has No Solution

Given the following system, find the solution:

x = y – 2

2x – 2y = 2

**Possible Answers:**

(1, 1)

no solution

(0, 1)

(1, 2)

(0, 0)

**Correct answer:**

no solution

When 2 equations in a system have the same slopes, they will either have no solution or infinite solutions. Since the y-intercepts are not the same, there is no solution to this system.

### Example Question #1 : How To Find Out When An Equation Has No Solution

Solve for :

**Possible Answers:**

No solution

Infinite Solutions

**Correct answer:**

No solution

Like other "solve for x" problems, to begin it, the goal is to get x by itself on one side of the equals sign. In this problem, before doing so, the imaginary -1 in front of (-27x+27) must be distributed.

Once this is done, you may start to try to get x by itself.

However, when subtracting 27x from either side and doing the same on the other, the 27x term cancels out. As a result, the equation becomes:

We know this is an untrue statement because these numbers are 5 spaces away from each other on the number line. The final answer is **No Solution**.

### Example Question #1 : How To Find Out When An Equation Has No Solution

Given the following system, find the solution:

**Possible Answers:**

No solution

**Correct answer:**

No solution

When two equations have the same slope, they will have either no solution or infinite solutions. By putting both equations into the form , we get:

and

With the equations in this form, we can see that they have the same slope, but different y-intercepts. Therefore, there is no solution to this system.

### Example Question #1 : How To Find Out When An Equation Has No Solution

Solve the following equation for :

**Possible Answers:**

Infinite solutions

No solution

**Correct answer:**

No solution

In order to solve for , we must get by itself on one side of the equation.

First, we can distribute the on the left side of the equal sign and the on the right side.

When we try to get by itself, the terms on each side of the equation cancel out, leaving us with:

We know this is an untrue statement, so there is no solution to this equation.

### Example Question #3 : Equations / Inequalities

Find the solution to the following equation if x = 3:

y = (4x^{2} - 2)/(9 - x^{2})

**Possible Answers:**

0

no possible solution

6

3

**Correct answer:**

no possible solution

Substituting 3 in for x, you will get 0 in the denominator of the fraction. It is not possible to have 0 be the denominator for a fraction so there is no possible solution to this equation.

### Example Question #1 : Linear / Rational / Variable Equations

**Possible Answers:**

3

1

–1/2

There is no solution

–3

**Correct answer:**

There is no solution

### Example Question #1 : How To Find Out When An Equation Has No Solution

**Possible Answers:**

None of the other answers

**Correct answer:**

A fraction is considered undefined when the denominator equals 0. Set the denominator equal to zero and solve for the variable.

### Example Question #1 : How To Find Out When An Equation Has No Solution

Solve for .

**Possible Answers:**

No solutions.

**Correct answer:**

No solutions.

Cross multiplying leaves , which is not possible.

### Example Question #1 : Equations / Inequalities

I. *x* = 0

II. *x* = –1

III. *x* = 1

**Possible Answers:**

II and III only

I only

I, II, and III

II only

III only

**Correct answer:**

I only

### Example Question #1 : Equations / Inequalities

Solve:

**Possible Answers:**

**Correct answer:**

First, distribute, making sure to watch for negatives.

Combine like terms.

Subtract 7x from both sides.

Add 18 on both sides and be careful adding integers.

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