SAT MAth Practice questions – all topics
- Algebra Weightage: 35% Questions: 13-15
- Linear equations in one variable
- Linear equations in two variables
- Linear functions
- Systems of two linear equations in two variables
- Linear inequalities in one or two variables
SAT MAth and English – full syllabus practice tests
Key Facts
1. An equation involving only one variable of degree 1 (power of the variable does not exceed 1) is called a linear equation.
2. The value of the variable that makes both the sides of the equation equal is called the solution or root of the equation.
3. The equation remains unchanged by
(i) adding the same number to both the sides of an equation or
(ii) subtracting the same number from both the sides of the equation or
(iii) multiplying both the sides of the equation by the same non-zero number or
(iv) dividing both the sides of the equation by the same non-zero number.
4. Transposing a term from LHS to RHS changes the sign of the term from (+ve) to (–ve) and (–ve) to (+ve).
5. To Solve a linear equation in one variable, take all the terms involving the variable on one side and the constant terms to the other. Reduce the equation to the from cx = d, where \(x=\frac{c}{d}\)
1-1. Variables and Expressions
In algebra, variables are symbols used to represent unspecified numbers or values. An algebraic expression is a collection of numbers, variables, operations, and grouping symbols.
Example 1 Write an algebraic expression for each verbal expression.
a. Ten less than one-fourth the cube of \(p\).
b. Twice the difference between \(x\) and sixteen.
c. Four times the sum of a number and three.
d. Four times a number increased by three.
▶️Answer/Explanation
Solution
a. \(\frac{1}{4} p^3-10\)
b. \(2(x-16)\)
c. \(4(n+3)\)
d. \(4 n+3\)
1-2. Exponents and Order of Operations
An expression like \(3^5\) is called a power. The number 3 is the base, and the number 5 is the exponent.
\[
3^5=\underbrace{3 \cdot 3 \cdot 3 \cdot 3 \cdot 3}_{5 \text { factors of } 3}
\]
To evaluate an expression involving more than one operation, we agree to perform operations in the following order.
Order of Operations
1. Simplify the expressions inside grouping symbols, such as parentheses, brackets, and fraction bars.
2. Evaluate all powers.
3. Do all multiplications and divisions in order from left to right.
4. Do all additions and subtractions in order from left to right.
Example 1 . Evaluate \(\left(11-20 \div \frac{5^2-13}{3}+8\right) \times 2\)
▶️Answer/Explanation
Solution
\[
\begin{aligned}
& \left(11-20 \div \frac{5^2-13}{3}+8\right) \times 2 \\
& =\left(11-20 \div \frac{25-13}{3}+8\right) \times 2 \quad \text{Evaluate power inside grouping symbols.}\\
& =\left(11-20 \div \frac{12}{3}+8\right) \times 2\quad \text{Evaluate expression inside grouping symbols.} \\
& =(11-20 \div 4+8) \times 2 2\quad \text{Evaluate expression inside grouping symbols.}\\
& =(11-5+8) \times 2 2\quad \text{Divide 20 by 4.}\\
& =(6+8) \times 2 2\quad \text{Subtract 5 from 11.} \\
& =(14) \times 2 2\quad \text{Evaluate expression inside grouping symbols.} \\
& =28 2\quad \text{Multiply.}
\end{aligned}
\]
1-3. Simplifying Algebraic Expressions
A term is a number, a variable, or a product or quotient of numbers and variables. For example \(5, x\), \(7 a, b^2\), and \(2 m^3 n\) are all terms. Like terms contain identical variables. For example, in \(5 x^2-3 x^2+3 x\), the terms \(5 x^2\) and \(-3 x^2\) are like terms because the variable part of each term is identical.
The coefficient of a term is a number that multiplies a variable. For example, in \(8 x^2 y\), the coefficient is 8 , and in \(\frac{4 m}{5}\), the coefficient is \(\frac{4}{5}\).
An expression is in simplest form when it is replaced by an equivalent expression having no like terms or parentheses. Simplifying means rewriting in simpler form.
Distributive Property
Symbols For any real numbers \(a, b\), and \(c\)
\(a(b+c)=a b+a c \quad a(b-c)=a b-a c\)
Examples \(\quad 4(7+3)=4 \cdot 7+4 \cdot 3 \quad 4(7-3)=4 \cdot 7-4 \cdot 3\)
Commutative Property
Symbols For any real numbers \(a\) and \(b\),
\(a+b=b+a \quad a \cdot b=b \cdot a\)
Examples \(3+4=4+3\)
\(3 \cdot 4=4 \cdot 3\)
Associative Property
Symbols For any real numbers \(a, b\), and \(c\)
\((a+b)+c=a+(b+c) \quad(a b) c=a(b c)\)
Examples \((3+4)+7=3+(4+7)\)
\((3 \cdot 4) \cdot 7=3\).
1-4. Rational, Irrational, and Decimal
Numbers can be pictured as points on a horizontal line called a number line. The point for 0 is the origin. Points to the left of 0 represent negative numbers, and points to the right of 0 represent positive numbers. Numbers increase in value from left to right. The point that corresponds to a number is called the graph of the number. Each number in a pair such as 3 and -3 is called the opposite of the other number. The opposite of \(a\) is written \(-a\).
natural numbers: \(\quad\{1,2,3, \ldots\}\)
whole numbers: \(\quad\{0,1,2,3, \ldots\}\)
integers: \(\quad\{\ldots,-3,-2,-1,0,1,2,3, \ldots\}\)
rational numbers: A rational number is one that can be expressed as a ratio \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b\) is not zero, such as \(-3.72,-\frac{2}{3}, 0,2\), and \(4 . \overline{23}\). The decimal form of a rational number is either a terminating or repeating decimal.
irrational numbers Any real number that is not rational is irrational. \(-\sqrt{3}, \sqrt{2}\), and \(\pi\) are irrational.
Rounding Decimals
To round a decimal to the desired place, underline the digit in the place to be rounded.
1) If the digit to the right of the underlined digit is 5 or more, increase the underlined digit by one (round up).
2) If the digit to the right of the underlined digit is less than 5 , leave the underlined digit as it is (round down).
3) Drop all digits to the right of the underlined digit.
2-1. Writing Equations
An equation is a mathematical sentence with an equal sign. To translate a word sentence into an equation, choose a variable to represent one of the unspecified numbers or measures in the sentence. This is called defining a variable. Then use the variable to write equations for the unspecified numbers.
Consecutive Numbers
Consecutive Integers \(\quad \ldots,-3,-2,-1,0,1,2,3, \ldots \quad n, n+1, n+2\) are three consecutive integers if \(n\) is an integer.
Consecutive Even Integers \(\quad \ldots,-6,-4,-2,0,2,4,6, \ldots \quad n, n+2, n+4\) are three consecutive even integers if \(n\) is an even integer.
Consecutive Odd Integers \(\quad \ldots,-5,-3,-1,1,3,5, \ldots \quad n, n+2, n+4\) are three consecutive odd integers if \(n\) is an odd integer.
Example 1 – Translate each sentence into an equation.
a. Twice a number increased by fourteen is identical to fifty. ▶️Answer/ExplanationSolution a. Let \(c\) be the number. Define a variable. \(\underbrace{2c}_{\text {Twice a number }} \underbrace{+}_{\text {increased by }} \underbrace{14}_{\text {fourteen }} \underbrace{=}_{\text {is identical to }} \underbrace{50}_{\text {fiffy. }}\) | b. Half the sum of seven and a number is the same as the number decreased by two. ▶️Answer/ExplanationSolution Let \(n\) be the number. Define a variable. \[ |
2-2. Solving Equations
To solve an equation means to find all values of the variable that make the equation a true statement. One way to do this is to isolate the variable that has a coefficient of 1 onto one side of the equation. You can do this using the rules of algebra called properties of equality.
\[
\begin{array}{lll}
\text { Properties of Equality } & \text { Symbols } & \text { Examples } \\
\text { 1. Addition Property } & \text { If } a=b \text {, then } a+c=b+c . & \text { If } x-3=5 \text {, then }(x-3)+3=(5)+3 . \\
\text { 2. Subtraction Property } & \text { If } a=b \text {, then } a-c=b-c . & \text { If } x+2=6 \text {, then }(x+2)-2=(6)-2 . \\
\text { 3. Multiplication Property } & \text { If } a=b \text {, then } c a=c b . & \text { If } \frac{1}{2} x=3 \text {, then } 2 \cdot \frac{1}{2} x=2 \cdot 3 . \\
\text { 4. Division Property } & \text { If } a=b \text { and } c \neq 0 \text {, then } \frac{a}{c}=\frac{b}{c} . & \text { If } 3 x=15 \text {, then } \frac{3 x}{3}=\frac{15}{3} .
\end{array}
\]
2-3. Solving Equations with Variables on Both Sides
Some equations have variables on both sides. To solve such equations, first use the Addition or Subtraction Property of Equality to write an equivalent equation that has all of the variables on one side. Then use the Multiplication or Division Property of Equality to simplify the equation if necessary. When solving equations that contain grouping symbols, use the Distributive Property to remove the grouping symbols.
Example 1 – Solve each equation.
a. \(\frac{7}{3} x-8=6+\frac{1}{3} x\)
▶️Answer/Explanation
Solution
Steps:
- Subtract \(\frac{1}{3} x\) from each side.
- Simplify.
- Add 8 to each side.
- Simplify.
- Divide each side by 2 .
- Simplify.
\[
\begin{aligned}
& \text { a. } \frac{7}{3} x-8=6+\frac{1}{3} x \\
& \frac{7}{3} x-8-\frac{1}{3} x=6+\frac{1}{3} x-\frac{1}{3} x \\
& 2 x-8=6 \\
& 2 x-8+8=6+8 \\
& 2 x=14 \\
& \frac{2 x}{2}=\frac{14}{2} \\
& x=7
\end{aligned}
\]
2-4. Equation with No Solution and Identity
It is possible that an equation may have no solution. That is, there is no value of the variable that will result in a true equation. It is also possible that an equation may be true for all values of the variable. Such an equation is called an identity.
Example 1
Solve each equation. If the equation has no solution or it is an identity, write no solution or identity.
a. \(2(1-x)+5 x=3(x+1)\) ▶️Answer/ExplanationSolution \[ The given equation is equivalent to the false statement \(2=3\). Therefore the equation has no solution. | b. \(5 w-3(1-w)=-2(3-w)\) ▶️Answer/ExplanationSolution \[ |
2-5. Solving for a Specific Variable
A formula is an equation that states the relationship between two or more variables. Formulas and some equations contain more than one variable. It is often useful to solve formulas for one of the variables.
Example 1 – Solve each equation or formula for the specified variable.
a. \(3 x-a=k x+b\), for \(x\)
▶️Answer/Explanation
Solution
Steps
- Subtract \(k x\) from each side.
- Simplify.
- Add \(a\) to each side.
- Simplify.
- Distributive property
- Divide each side by \(3-k\).
- Simplify.
\[
\begin{aligned}
& \text { a. } 3 x-a=k x+b \\
& 3 x-a-k x=k x+b-k x \\
& 3 x-a-k x=b \\
& 3 x-a-k x+a=b+a \\
& 3 x-k x=b+a \\
& x(3-k)=b+a \\
& \frac{x(3-k)}{3-k}=\frac{b+a}{3-k} \\
& x=\frac{b+a}{3-k} \\
&
\end{aligned}
\]