Semester Exam Review Part 3

Semester Exam Review Part 3

Systems of Equations

• 2 Equations, 3 types of Solutions

• 1 Solution (1 Intersection)

• No Solutions (No Intersection; Parallel)

• Infinite Solutions (Equal Lines; Overlap)

• Solutions will always be a coordinated, (x, y)

• Use 'Desmos' to find solutions, graph

Multiple Choice

Find the point of intersection of the lines in the figure, given line A has the equation y = x + 5 and 6x + 2y = 22. Write your answer as a fraction.

• x = 7/2, y = 9/2

• x = 9/2, y = 15/2

• x = 3/2, y = 13/2

• x = 5/2, y = 9/2

Multiple Choice

Use the substitution method to solve the system.

 $-2x+y=-2$  
 $8x-2y=16$  

• x = 6, y = 1

• x = 3, y = 4

• x = 2, y = 8

• x = 1, y = -3

Multiple Choice

 $4x\ -8y=20$  

 $-10x+20y=-50$  

How many solutions are there to this system?

• None

• Exactly 1

• Exactly 2

• Infinitely Many

Writing Infinitely Many Answers

• x will always be equal to x

• Solve for y

• Use the equation that has a negative x

• Step 1: Add x

• Step 2: Divide by number in front of y

Multiple Choice

Write the answer below as infinitely many:

 $4x-8y=20$  
 $-10x+20y=-50$  

•  $y=10x-50$  

•  $y=\frac{20}{10x-50}$  

•  $y=\frac{50-10x}{20}$  

•  $y=\frac{10x-50}{20}$  

Properties of Exponents

• When multiplying terms: Exponents Add, Numbers Multiply

• When dividing terms: Exponents Subtract, Numbers Divide

Multiple Choice

Divide using the quotient rule:

 $\frac{16a^{15}b^{10}c^9}{4a^{10}b^3c^6}$  

•  $4a^5b^7c^3$  

•  $12a^5b^7c^3$  

•  $4a^{1.5}b^{3.3}c^{1.5}$  

•  $12a^{25}b^{13}c^{15}$  

Properties of Exponents

• When you have exponents raised to other exponents

• Numbers get raised to the exponents

• Exponents Multiply

• Negative Exponents move to the bottom of a fraction and turn positive

Multiple Choice

 $\left(5a^7b^{-4}\right)^{-2}$  

Simplify. Do not leave negative exponents in your answer.

•  $25a^{14}b^8$  

•  $\frac{b^8}{25a^{14}}$  

•  $\frac{b^8}{10a^{14}}$  

•  $10a^{14}b^8$  

Adding and Subtracting Expressions

• Only Add and Subtract LIKE TERMS (Same variable, same exponent)

• If you are subtracting, remember to change signs. A negative is distributed

Multiple Choice

Subtract:

 $\left(6t^6-4t^5-3t^2-8\right)-\left(-4t^6+3t^5-2t^2-10\right)$ 

•  $2t^6-t^5-5t^2-18$  

•  $2t^6-7t^5-5t^2-2$  

•  $10t^6-7t^5-t^2+2$  

•  $10t^6-t^5-5t^2-18$  

Evaluating Expressions

• Substitute the given value

• Follow Order of Operations

• PEMDAS

Multiple Choice

 Evaluate the following algebraic expression:$x^2-x+11\ for\ x\ =\ 5$  

• 25

• 16

• 18

• 31

Multiplying Polynomials

• Know the size of the polynomials

• Monomial (1), Binomial (2), Trinomial (3)

• Construct a box to multiply polynomials

• Combine Like Terms

• Remember

   $x\cdot x=x^2\ and\ x\cdot x^2=x^3$  

Squaring a Binomial

• A binomial that is multiplied to itself

• 2 x 2

• Use Box Method

Multiple Choice

Multiply the polynomials:

 $\left(x-3\right)\left(x^2+5x-5\right)$  

•  $x^3+2x^2-20x+15$  

•  $x^3+7x^2-11x^2+15$  

•  $x^3+8x^2-2x+15$  

•  $x^3-2x+8x+15$  

Multiple Choice

Square the binomial:

 $\left(2x-6\right)^2$  

•  $4x^2+36$  

•  $12x^2$  

•  $4x^2-16x-36$  

•  $4x^2-24x+36$  

Simplifying Rational Expressions

• Factor out terms

• Cancel Out Terms from the Numerator (Top) with the Denominator (Bottom)

• If Dividing a 2nd Fraction remember to flip the 2nd Fraction and multiply it.

Multiple Choice

Simplify the rational expression:

 $\frac{x^2-3x-10}{x^2-9x+20}$  

•  $\frac{x+2}{x-4}$  

•  $\frac{x-2}{x+4}$  

•  $\frac{x+2}{x+4}$  

•  $\frac{x-4}{x+2}$  

Multiple Choice

Divide and simplify your answer:

 $\frac{x^2-49}{x^2-4x-21}\div\frac{x}{x+3}$  

•  $\frac{x-7}{x}$  

•  $\frac{x}{x-7}$  

•  $\frac{x+7}{x}$  

•  $\frac{x}{x+7}$