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Linear equations on the SAT test

Linear equations on the SAT test

SAT Subscore: Heart of Algebra

Linear equations formula sheet

linear equation is an algebraic equation in which each term has an exponent of one. It is called linear because it can be graphed as a straight line in the xy-plane. 

On the SAT test linear equations are part of heart of algebra subscore questions. Since this is a fundamental topic it appears in many SAT questions. 

Let’s look at the different types of questions and examples for each one of them.

Calculating an output of an expression SAT questions

In these questions you need to calculate the value of an expression. In some questions you are given a word problem and in other questions you are given an equation that you have to use it order to solve the expression.

The skills required are
Translating a word problem into a mathematical expression.
Solving equation.
Performing calculations on an equation to match a given expression.

Before making any calculations, it is necessary to know the rules for calculations with negative numbers:
Negative * negative = positive. For example: -5*-4=20
Positive * negative = negative. For example: 5*-4=-20
Negative : negative = positive. For example: -8:-4=2
Positive : negative = negative. For example: 8:-4=-2
Negative : positive = negative. For example: -8:4=-2

Calculating an output of an expression given a word problem

Consider this example:

The length of the rectangle is 5 cm, the width of the rectangle is 20 percent smaller than its length.

What is the area of the rectangle?

We need to write an expression and calculate it. Rectangle’s area is the equals to its length multiplied by its width. If the length of a rectangle is 5 cm then the width of the rectangle is 5* 80%=4 cm. The area of ​​the rectangle is 5*4=20 cm.

Calculating and expression given an equation

In these questions we are be asked to calculate an expression given an equation that we don’t need to solve. All we need in order to get the answer is to change the equation to match the expression. 

Consider this example:

2x+10=3x+25

What is 6x+50 equal to?

Remember that if you need to divide or multiple you need to do it for each component of the expression. In this case you can add parentheses to the expression. We see that the expression 6X+50 is 2 times larger than the expression 3x+25 therefore we can multiply the given equation by 2 and get the answer.

Before multiplying each side of the equation by 2 we add parentheses to the both sides of the equation:
(2x+10)*2=(3x+25)*2
4x+20=6x+50
So the answer is that 6x+50 is equal to 4x+20

Sometimes we will be asked to calculate an expression given an equation that we need to solve first. 

Consider this example:

We know that 3x+10=25, calculate 4x+12.

Step 1: Solving the equation:
3x+10=25
3x=25-10
3x=15
x=5

Step 2: Calculating the expression:
4x+12=4*5+12=20+12=32

Solving a linear equation SAT questions

In these questions you are given an equation and you are asked to solve it.

The skills required are solving an equation with one variable and making calculations including fractions, decimal fractions and absolute values.
In addition there are questions with two variables where you need to find the value of one variable from a word problem before solving for the second variable.

The rules for maintaining equality

Before starting to solve the equation, it is necessary to know the rules for maintaining equality:
1. You can add or subtract the same value from each side of the equation.
2. You can divide or multiply each side of the equation by a same value.

Consider this example:

2x+5=14

What is the solution to the given equation?

2x+5=14
2x+5-5=14-5
2x=9
2x:2=9:2
x=4.5

 

The steps for solving a linear equation

Distributing formula for removing parentheses:

(a+b)x=ax+bx
(a-b)x=ax-bx

For example: (2+x)5=10+5x

Combining like terms formula:

ax+bx=(a+b)x
ax-bx=(a-b)x

For example: 8x-2x=(8-2)x=6x

The goal of solving an equation is finding the x variable value. In order to do so we need to isolate x on the one side of the equation so that the other side of the equation will contain the constant (the answer).

a. Removing parentheses and combining like terms in each side of the equation.
Distributing formula for removing parentheses: (a±b)x=ax±bx, for example (2+x)5=10+5x
Combining like terms formula: ax±bx=(a±b)x, for example 8x-2x=(8-2)x=6x
b. Isolating the variable term on one side of the equation by using addition or subtraction.
c. Solving the equation (finding x=) by using multiplication or division.

Consider this example:

5x-10+x=2x+9(5+5)

What is the solution to the given equation?

Step 1: Solving the equation
a. Removing parentheses and combining like term in each side of the equation:
6x-10=2x+90

b. Subtracting 2x and adding 10 to each side of the equation:
6x-10-2x+10=2x+90-2x+10
4x=100

c. Dividing each side of the equation by 4:
4x:4=100:4
x=25

Step 2: checking the answer:
5*25-10+25=2*25+9(5+5)
125-10+25=50+9*10
140=140

Solving a linear equation with decimal fractions

Consider this example:

(2x-6)*0.5=(3x-10)*0.4-0.6

What is the solution to the given equation?

Step 1: Solving the equation:
To solve the equation we need to open parentheses on both of its sides.
2*0.5x-6*0.5=3*0.4X-10*0.4-0.6
x-3=1.2x-4-0.6
-0.2x=-4-0.6+3
-0.2x=-1.6
x=1.6/0.2
x=8

Step 2: checking the answer:
(2*8-6)*0.5=10*0.5=5
(3*8-10)*0.4-0.6=14*0.4-0.6=5.6-0.6=5

Solving a linear equation with fractions

Addition and subtraction formula:

 ab±cb=(a±c)b

For example: 14+514=512

Multiplication formula:

ab*cd=a*cb*d

For example: 14*2=1*24*1=24 =12

Division formula:

ab : cd=ab*dc=a*db*c

For example: 14 : 2=14*12=1*14*2=1

To add or subtract fractions they must have the same denominator (the bottom value). We need to add or subtract the numerators and leave the denominator.
Addition and subtraction formula: ab±cb=(a±c)b.   For example: 14+514=512

To multiply fractions, you have to multiply the nominators and multiply the denominators.
Multiplication formula: ab*cd=a*cb*d. For example: 14*2=1*24*1=24 =12

To divide fractions, you must flip the second fraction and then multiply it with the first fraction.
Division formula: ab : cd=ab*dc=a*db*c. For example: 14 : 2=14*12=1*14*2=1

If the denominators are different you can first reach a common denominator in order to cancel the denominators and then you can continue to solve the equation. This is done by multiplying by the smallest common multiple of the denominators of the fractions.

Consider this example:

312x-214=2x+514

What is the solution to the given equation?

Step 1: Solving the equation: The common denominator for the numbers 2 and 4 is 4 so we need to multiply the equation by 4. 4*(312x-21⁄4)=4*(2x+51⁄4) 4*3x +4*12x-4*2-4*1⁄4=8x+4*5+ 4*1⁄4 12x+2x-8-1=8x+20+1 14x-9=8x+21 6x=30 x=5 Step 2: Checking the solution: 312x-21⁄4=2x+51⁄4 312*5-21⁄4=2*5+514 15+2+12-21⁄4=1514 17+12-21⁄4=1514 1514=1514

Solving equations with absolute values

Pay attention that absolute value calculations are rare on the SAT.

The absolute value of a negative number is always positive. Therefore, we need to split an absolute value equation into 2 equations.

Consider this example:

What is the solution for the equation |6x-10|=2?

Step 1: Solving the equation
Since we don’t know the sign of |6x-10|=2, we need to write 2 equations.

Equation 1: (6x-10)=2
6x=12
x=2

Equation 2: (6x-10)=-2
6x=8
x=86=126=113

Step 2: Checking the answers
|6*2-10|=2
2=2

|6*86-10|=2
|486-10|=2
|8-10|=2
|-2|=2

Solving an equation with two variables given an input of one variable

In these questions you are given a word problem and an equation with two variables. You need to find the value of one variable inside the word problem and then plug it into the equation. After that you will have an equation with one variable that you can solve.

Consider this example:

0.6x+0.4y=86

The equation above represents a weighted average of 86  in two subjects, defined x as a grade in course 1 and y as a grade in course 2. What was the second grade?

The first grade should be placed in the equation instead of x and then we will be able to solve an equation with 1 variable y.

Step 1: Plugging the input into the equation:
0.6*90+0.4y=86

Step 2: Solving the equation with 1 variable:
54+0.4y=86
0.4y=32
y=80
The second grade was 80.

Step 3: Checking the answer:
In order to test the answer we will place it in the equation: 0.6*90+0.4*80=54+32=86.

Linear equations that don't have one solution SAT questions

Previous examples included common equations that had one single solution.

In these questions you are asked to solve an equation that has no solution or an infinite number of solutions.

The skills required are solving an equation with one variable and identifying the conditions for no solution or an infinite number of solutions to the equation.

Linear equations with no solution

Equations with no solution appear when the x variable is eliminated from the equation so what is left is only 2 constants a=b. Since a is a different from b this is a false statement and the equation has no solution. 

Consider this example:

What is the solution for the equation 6x+12=3(2x+2)?

6x+12=3(2x+2)
6x+12=6x+6
12-6=6x-6x
6=0 is a false statement therefore the equation has no solution.

Finding the condition for no solution:

In this type of questions we are given an equation with a as a constant and asked for which value of a there is no solution to the equation. In order to find the solution, we need to solve the equation. Another possibility is to place the answers into the equation and see if we get a false statement.

Consider this example:

For which value of a the equation 6ax+12=3(2x+2) has no solution? a=0,1,2?

Option 1: Solving the equation
6ax+12=3(2x+2)
6ax+12=6x+6
6ax-6x=-6
6x(a-1)=-6
x(a-1)=-1
x=-1/(a-1)
a≠1 since division by zero is not allowed therefore there is no solution when a=1.

Option 2: Placing given answers into the equation
For a=0 we will get
12=6x+6
6x=6
x=1
We can check this answer by plugging x=1 and a=0 into the equation to see if the quality holds:
6*0*1+12=3(2*1+2)
0+12=3*4
12=12

For a=1 we will get
6x+12=6x+4
12=4 is a false statement therefore there is no solution when a=1.

For a=2 we will get
12x+12=6x+6
6x=-6
x=-1
We can check this answer by plugging x=-1 and a=2 into the equation to see if the quality holds:
6*2*-1+12=3(2*-1+2)
-12+12=3*(-2+2)
0=3*0
0=0

Linear equations with an infinite number of solutions

Equation with an infinite number of solutions appears when the constants are eliminated from the equation so what is left is only 2 x variables in the form x=x.  This is a true statement for every x therefore the equation has an infinite number of solutions. 

Consider this example:

What is the solution for the equation 6x+12=3(2x+4)?

6x+12=3(2x+4)
6x+12=6x+12
6x=6x
x=x
x=x is a true statement for every x therefore the equation has an infinite number of solutions.

We can also continue to solve:
x=x
x-x=0
0=0 is also a true statement for every x

Creating and solving an equation from a word problem SAT questions

In these questions you are given a word problem from which you need to write an equation and solve it.  

The skills required are defining x variable in a word problem, writing an equation from a word problem and solving an equation with one variable.

Consider this example:

The inventory of a store contained bags with balls received from the supplier so that in each bag were packed 10 balls. The store worker decided to pack the balls in smaller packages in order to sell them to customers. He took a certain number of bags and emptied the balls from them. He then added to the balls he took out another 30 balls left in the store from previous orders and then he divided these balls into 75 bags so that each bag contained 2 balls. The worker noticed that he forgot to write down how many bags he had taken from stock. What was the amount?

In this example we will need to write an equation in order to reach a solution.

Step 1: X definition: Let’s look at the question and mark in x the number of bags the employee took from stock.
The worker took x bags containing 10 balls in each bag and added them 30 balls: 10x+30
Then he divided these the balls into 75 bags so that each bag contained 2 balls: 75*2

Step 2: writing the expressions in each side of equation: We need to find 2 equal expressions.
Since we know that the worker sorted the same amount of balls, we know what to write on each side of the equation:
10x+30=75*2.

Step 3: Solving the equation:
10x+30=75*2
10x=150-30
10x=120
x=12
The solution is x=12.

Solving a word problem using two equations

What if the question is not about is the x variable value? In this case after calculating x value, we will write another equation using y variable (y will be defined as what is asked in the question).

Steps for writing a second equation: 

Step 1: Looking at the relevant data parts again, this time ignoring the data we needed to calculate x.
Step 2: Identifying parts of the data which will give us an equality.
Step 1+2: Writing the second equation consisting y variable and solving it.

Let’s add to the above example: There were 50 bags with balls in the store’s inventory.
I addition, let’s assume that the question is not to find the amount off bags that the worker took from the store’s inventory. Instead, the question is how many bags were left in the inventory?

The inventory of a store contained bags with balls received from the supplier so that in each bag were packed 10 balls. There were 50 bags with balls in the store’s inventory. The store worker decided to pack the balls in smaller packages in order to sell them to customers. He took a certain number of bags (we solved x=12)  and emptied the balls from them. He then added to the balls he took out another 30 balls left in the store from previous orders and then he divided these balls into 75 bags so that each bag contained 2 balls. How many bags were left in the inventory?

The first equation was 10x+30=75*2. We solved it so x=12.

Writing a second equation: We will use the letter y for the new variable. The y variable will be according to the question: y=the number off bags that were in the inventory. The equation will be: y=50-x. We had already solved x=12, so now we know that y=38.

The question could also be even more complicated: How many bags were left in the inventory? In this case the second equation should be y=(50-12)*10 and the answer will be 380 balls.

Linear equations formula sheet

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