Linear / Rational / Variable Equations - ACT Math
Card 0 of 20
I. x = 0
II. x = –1
III. x = 1
I. x = 0
II. x = –1
III. x = 1
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Given the following system, find the solution:
x = y – 2
2x – 2y = 2
Given the following system, find the solution:
x = y – 2
2x – 2y = 2
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.
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.
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Find the solution to the following equation if x = 3:
y = (4x2 - 2)/(9 - x2)
Find the solution to the following equation if x = 3:
y = (4x2 - 2)/(9 - x2)
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.
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.
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A fraction is considered undefined when the denominator equals 0. Set the denominator equal to zero and solve for the variable.
A fraction is considered undefined when the denominator equals 0. Set the denominator equal to zero and solve for the variable.
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Solve for 
.
Cross multiplying leaves 
, which is not possible.
Cross multiplying leaves
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Solve:
Solve:
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.
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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Solve:
Solve:
First, distribute the 
to the terms inside the parentheses.
Add 6x to both sides.
This is false for any value of 
. Thus, there is no solution.
First, distribute the
Add 6x to both sides.
This is false for any value of
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Solve for 
:
Solve for
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.
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.
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Given the following system, find the solution:
Given the following system, find the 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.
When two equations have the same slope, they will have either no solution or infinite solutions. By putting both equations into the form
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.
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Solve the following equation for 
:
Solve the following equation for
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.
In order to solve for
First, we can distribute the
When we try to get
We know this is an untrue statement, so there is no solution to this equation.
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Solve 
.
Solve
By definition, the absolute value of an expression can never be less than 0. Therefore, there are no solutions to the above expression.
By definition, the absolute value of an expression can never be less than 0. Therefore, there are no solutions to the above expression.
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In the equation below, 
, 
, and 
are non-zero numbers. What is the value of 
in terms of 
and 
?
In the equation below,
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John, Jill, and Jack are splitting a pizza. John eats 
of the pizza and Jill eats 
of the pizza. How much of the pizza is left for Jack?
John, Jill, and Jack are splitting a pizza. John eats
We can write an equation for the amount of pizza eaten, with 
as the amount left for Jack.
To solve this equation, we must find the lowest common denominator of 
and 
. We can list the multiples of 
and 
to find the least common multiple:
We can see that the least common multiple of 
and 
is 
, so we can rewrite each of the fractions with a denominator of 
.
When we put these fractions back into the equation, we can solve for 
:
We can write an equation for the amount of pizza eaten, with
To solve this equation, we must find the lowest common denominator of
We can see that the least common multiple of
When we put these fractions back into the equation, we can solve for
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What is the slope of a line represented by the equation:
What is the slope of a line represented by the equation:
To solve problems where you need to find the slope of a line in a given equation, change the equation so that it matches y-intercept form:
For this equation, first move the 3x over to the other side of the equation.
The equation should now look like this:
Then, divide by 15 to isolate the variable 
.
Then simplify
Whatever number is before the x in the equation (m) is your slope.
To solve problems where you need to find the slope of a line in a given equation, change the equation so that it matches y-intercept form:
For this equation, first move the 3x over to the other side of the equation.
The equation should now look like this:
Then, divide by 15 to isolate the variable
Then simplify
Whatever number is before the x in the equation (m) is your slope.
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What is the value of (5 + x)(10 – y) when x = 3 and y = –3?
What is the value of (5 + x)(10 – y) when x = 3 and y = –3?
This is a simple plug-in and PEMDAS problem. First, plug in x = 3 and y = –3 into the x and y. You should follow the orders of operation and compute what is within the parentheses first and then find the product. This gives 8 * 13 = 104. The answer is 104.
This is a simple plug-in and PEMDAS problem. First, plug in x = 3 and y = –3 into the x and y. You should follow the orders of operation and compute what is within the parentheses first and then find the product. This gives 8 * 13 = 104. The answer is 104.
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If x = 4, and y = 3x + 5, then 2y – 1 equals
If x = 4, and y = 3x + 5, then 2y – 1 equals
Start by plugging in x = 4 to solve for y: y = 3 * 4 + 5 = 17. Then 2 * 17 – 1 = 33
Start by plugging in x = 4 to solve for y: y = 3 * 4 + 5 = 17. Then 2 * 17 – 1 = 33
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Sarah’s current age is three times Ron’s age two years ago. Sarah is currently 14 years older than Ron. What is the sum of Sarah and Ron’s current age?
Sarah’s current age is three times Ron’s age two years ago. Sarah is currently 14 years older than Ron. What is the sum of Sarah and Ron’s current age?
The best way to solve this problem is to turn the two statements into equations calling Sarah’s age S and Ron’s age R. So, S = 3(R – 2) and S = 14 + R. Now substitute the value for S in the second equation for the value of S in the first equation to get 14 + R = 3(R – 2) and solve for R. So R equals 10 so S equals 24 and the sum of 10 and 24 is 34.
The best way to solve this problem is to turn the two statements into equations calling Sarah’s age S and Ron’s age R. So, S = 3(R – 2) and S = 14 + R. Now substitute the value for S in the second equation for the value of S in the first equation to get 14 + R = 3(R – 2) and solve for R. So R equals 10 so S equals 24 and the sum of 10 and 24 is 34.
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