Math Reviewer PDF

MATH REVIEWER 1^st^ Quarter **[Factoring Polynomials]** **Factoring completely different types of polynomials** **Factoring** \- a mathematical process of finding the expressions which when multiplied will result to the given result. \- complete factorization is having all factors prime. \- a polynomial is prime when its factors are only itself and one. **[Common Monomial Factors]** **Greatest Common Factor (GCF)** \- largest quantity that is a factor of all the integers or polynomials involved. **Finding the GCF of the list of integers or terms** 1\. Prime factor the numbers 2\. Identify the common prime factors 3\. Take the product of all the common prime factors. If there are no common prime factors, the GCF is 1 **Factoring Polynomials** The first step in factoring a polynomial is to find the GCF of all its terms. Then we write the polynomials as a product by factoring out the GCF from all the terms. The remaining factors in each term will form a polynomial. **Example:** [9*x*^4 ^*y*^2^ − 15*x*^3^*y* + 3*x*^2^*y*]{.math.inline} GCF = [3*x*^2^*y*]{.math.inline} = [\$\\frac{{9x}\^{4}y\^{2}}{{3x}\^{2}y} - \\frac{{- 15x}\^{3}y}{{3x}\^{2}y} + \\frac{{3x}\^{2}y}{{3x}\^{2}y}\$]{.math.inline} = [3*x*^2^*y* − 5*x* + 1]{.math.inline} = [3*x*^2^*y*(3*x*^2^*y* − 5*x* + 1)]{.math.inline} **Note:** To check if correct; use distribution method **[Difference of Two Squares]** a binomial is the difference of two square if: 1\. both terms are squares. 2\. the sign of items are different **Formula:** [*a*^2^ − *b*^2^ = (*a* + *b*)(*a* − *b*)]{.math.inline} Factoring the difference of two squares, means obtaining the sum and difference of their square roots. **Factor:** [*x*^2^ − 4*y*^2^]{.math.inline} = [(*x*)^2^ − (2*y*)^2^←]{.math.inline} **Difference of two squares** = [(*x*−2*y*)(*x* + 2*y*)]{.math.inline} **Examples:** [\$\\frac{r\^{2}}{16} - 25\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ {- c}\^{4} + d\^{4}\$]{.math.inline} = [\$(\\frac{r}{4})\^{2}\\left( 5 \\right)\^{2}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ = (d\^{2} + c\^{2})(d + c)(d - c)\$]{.math.inline} = [\$(\\frac{r}{4} - 5)(\\frac{r}{4} + 5)\$]{.math.inline} **[Sum or Difference of Two Cubes]** **Perfect Cubes** something times something times something. Where the something is a factor 3 times. 8 is (2) (2) (2), so 8 is a perfect cube. x^6^ is (x^2^) (x^2^) (x^2^), so x^6^ is a perfect cube. Is it easy to see if a variable is a perfect cube. See if the exponent is divisible by 3. **Perfect Cube and Cube Roots:** 1^3^ = 1 6^3^ = 216 2^3^ = 8 7^3^ = 343 3^3^ = 27 8^3^ = 512 4^3^ = 64 9^3^ = 729 5^3^ = 125 10^3^ = 1000 The sum or difference of two cubes will factor into a binomial x trinomial. **Procedures:** Get the cube roots of the given and make them into a binomial with the sign the same as for the given cubes. Square the cube roots and make them the first and third terms of the trinomial. Multiply the cube roots and make the product as the second term of the trinomial with the sign opposite to the given cubes. Always Opposite Sign [ ↓ ↓]{.math.inline} [*a*^3^ + *b*^3^ = (*a* + *b*)(*a*^2^ − *ab* + *b*^2^)]{.math.inline} [ ↑]{.math.inline} [ ↑]{.math.inline} [↑]{.math.inline} Same Sign Always Positive Square these terms [ ↓ ↓ ↓ ↓]{.math.inline} [*a*^3^ + *b*^3^ = (*a* + *b*)(*a*^2^ − *ab* + *b*^2^)]{.math.inline} [ ↑ ↑ ↑]{.math.inline} Cube root of 1^st^ term Product of cube root Cube root of 2^nd^ term of 1^st^ and 2^nd^ term **Example:** [27*x*^2^ − 125 = (3*x* − 5)(9*x*^2^ + 15*x* + 25]{.math.inline} **Note:** Always check for the GCF first no matter what **Expression with Parenthesis:** [(2*x* − 3)^2^ − 1]{.math.inline} = [((2*x*−3) − 1)(2*x* − 3)^2^ + ((2*x*−3) + 1)]{.math.inline} = [(2*x* − 4)(4*x*^2^ − 12*x* + 9 + 2*x* − 3 + 1)]{.math.inline} = [(2*x*−4)(4*x*^2^−10*x*+7)]{.math.inline} = [2(*x* − 2)(4*x*^2^ − 10*x* + 7)]{.math.inline} **[Factoring Perfect Square Trinomials]** Twice the product of first and last term Last term [ ↓ ↓]{.math.inline} [*a*^2^ + 2*ab* + *b* = (*a*+*b*)(*a*−*b*) = (*a* + *b*)^2^]{.math.inline} [↑ ↑ ↑]{.math.inline} Square of Square of 1^st^ term 1^st^ term Last term **Factor:** [*m*^2^ + 12*m* + 36 = (*m* + 6)^2^]{.math.inline} **Note:** 1^st^ sign is the factor sign **Square of Binomials** **Procedure:** Square 1^st^ term Twice the product of 1^st^ and last term Square last term **Example:** [(*x* + 5)^2^ = *x*^2^ + 10*x* + *b*^2^]{.math.inline} **[Factoring General Trinomials]** **Case 1** **Step 1**: List all pairs of numbers that when multiplied it will result to the last term. **Step 2**: Choose the pair that adds up to the middle coefficient. **Step 3**: Fill the numbers into the blanks in the binomial. **Example:** [*x*^2^ + 7*x* + 12]{.math.inline} **Step 1:** 12 = 1 12 = 2 6 = 3 4 **Step 2:** 3 + 4 = 7 **Step 3:** [( *x*+3 )( *x* + 4 )]{.math.inline} **Note:** If the terms are negative, list down the negative pairs. **Case 2** **Step 1:** Multiply the leading coefficient and constant. **Step 2:** List all the pairs of numbers that multiply to equal that product **Step 3:** Which adds up to the middle coefficient? **Step 4:** Write temporary factors with two numbers. **Step 5:** Put the original leading coefficient under both numbers. **Step 6:** Reduce fractions if possible. **Step 7:** Move denominators in front of variable. **Example:** [3*x*^2^ + 14*x*+ 8]{.math.inline} **Step 1:** 3 8 = 24 **Step 2:** 24 = 1 24 = 2 12 = 3 8 = 4 6 **Step 3:** 2 + 12 = 14 **Step 4:** [( *x*+ 2 )( *x*+12 )]{.math.inline} **Step 5:** [\$\\left( \\ x + \\ \\frac{2}{3} \\right)\\left( \\ x + \\ \\frac{12}{3}\\ \\right)\$]{.math.inline} **Step 6:** [\$\\left( \\ x + \\ \\frac{2}{3}\\ \\right)\\left( \\ x + 4\\ \\right)\$]{.math.inline} **Step 7:** [( 3*x*+2 )( *x* + 4 )]{.math.inline} **[Factoring by Group]** When polynomials contain four terms, it is sometimes easier to group like terms in order to factor. Your goal is to create a common factor. You can also move terms around in the polynomial to create a common factor. Practice makes you better in recognizing common factors. **Example:** **Factor:** [3*xy* − 21*y* + 5*x* − 25]{.math.inline} **Factor the first two terms:** [3*xy* − 21*y* = 3*y* (*x* − 7)]{.math.inline} **Factor the last two terms:** [+ 5*x* − 25 = 5 (*x* − 7)]{.math.inline} The parenthesis are the same so it's the common factor **Common Factor:** [( *x*−7 )( 3*y* + 5 )]{.math.inline} **If Opposite Signs:** **Factor:** [15*x* − 3*xy* + 4*y* − 20]{.math.inline} **Factor the first two terms:** [15*x* − 3*xy* = 3*x*(5−*y*)]{.math.inline} **Factor the last two terms:** [ + 4*y* − 20 = 4(*y*−5)]{.math.inline} The parenthesis are opposites so change the sign on the 4: -4 (-y+5) or -4 (5-y) **Common Factor:** [( 5−*y* )( 3*x* − 4 )]{.math.inline} **Types of Factoring** 1\. Look for GCF first. 2\. Count the number of terms: 4 terms \- factor by grouping 3 terms \- look for perfect square trinomial \- if not, quadratic trinomials 2 terms \- look for difference of squares \- or sum or difference of cubes If any ( ) still has an exponent of 2 or more, see if you can factor again. **[Word Problems Solved by Factoring\ ]** **Zero-Product Property** - states that if the product of two real numbers a and b is zero, then a = 0 or b = 0 or both a and b equal 0. **Quadratic equation** - an equation in the second degree and in type ax2 + bx + c = 0, where a is nonzero number. **Examples:** **Solve** [*x*^2^ + 7*x* + 6 = 0]{.math.inline} Quadratic equation → factor the left-hand side (LHS) [*x*^2^ + 7*x* + 6 = (*x* + 6)(*x* + 1)]{.math.inline} [*x*^2^ + 7*x* + 6 = (*x*+6)(*x*+1) = 0]{.math.inline} Now the equation as given is of the form ab = 0 Set each factor equal to 0 and solve [*x* + 6 = 0]{.math.inline} [*x* = − 6]{.math.inline} **Solve** [(4*t*+1)(3*t*−5) = 0]{.math.inline} Notice the equation as given is of the form ab = 0 Set each factor equal to 0 and solve [4*t* + 1 = 0 ]{.math.inline} Subtract 1 [3*t* − 5 = 0]{.math.inline} Add 5 [4*t* = − 1]{.math.inline} Divide by 4 [3*t* = 5]{.math.inline} Divide by 3 [\$t = - \\frac{1}{4}\$]{.math.inline} [\$t = \\frac{5}{3}\$]{.math.inline} Solution: [\$t = - \\frac{1}{4}\\text{and}\\frac{5}{3}\$]{.math.inline} [\$t = \\{ - \\frac{1}{4},\\frac{5}{3}\$]{.math.inline} **Methods for Problem Solving** **Understanding the problem** Read the problem very Understanding carefully. It may be necessary to read it several times. A sketch may help. **Devising a plan** Determine what to find, and then let a variable represent the unknown. It will be necessary to write an equation based upon the words of the problem. **Carrying out the plan** Solve the Equation **Looking Back** Check your answers. **Example 1** The product of one more than a number and 4 less than a number is 36. Find the number. [(*x*+1)(*x*−4) = 36]{.math.inline} [*x*^2^ − 3*x* − 4 = 36]{.math.inline} [*x*^2^ − 3*x* − 4 − 36 = 36 − 36]{.math.inline} [*x*^2^ − 3*x* − 40 = 0]{.math.inline} [(*x*−8)(*x*+5) = 0]{.math.inline} [*x* − 8 + 8 = 0 + 8 *x* + 5 − 5 = 0 − 5]{.math.inline} [*x* = 8 *or* *x* = − 5]{.math.inline} To Check: [(*x*+1)(*x*−4) = 36 (*x*+1)(*x*−4) = 36]{.math.inline} (8 + 1) (8 - 4) = 36 (-5 + 1) (-5 - 4) = 36 \(9) (4) = 36 ✓ (-4)(-9) = 36 ✓ **Example 2** The length of a rectangle is two feet less than 3 times the width. If the area is 65 ft^2^, find the dimensions. Let x = width = 5 3x-2 = length = 13 [*x*(3*x* − 2) = 65]{.math.inline} A = l w [65 = (3*x* − 2)*x*]{.math.inline} [65 = 3*x*^2^ − 2*x*]{.math.inline} [65 − 65 = 3*x*^2^ − 2*x* − 65]{.math.inline} [0 = 3*x*^2^ − 2*x* − 65]{.math.inline} [0 = (3*x* + 13)(*x* − 5)]{.math.inline} [*x* + 13 = 0 − 13 *x* − 5 = 0 + 5]{.math.inline} [*x* = − 13 *x* = 5]{.math.inline} To Check: [*x*(3*x* − 2) = 65]{.math.inline} [5\[3(5) − 2\] = 65]{.math.inline} [5(15 − 2) = 65]{.math.inline} [5(13) = 65]{.math.inline}✓ **Example 3** The product of two consecutive integers is 90. Find the integers. Let x = 1st integer = -10 or 9 x+1 = 2nd integer = -9 o 10 [*x*(*x*+1) = 90]{.math.inline} [*x*^2^ + *x* = 90]{.math.inline} [*x*^2^ + *x* − 90 = 90 − 90]{.math.inline} [*x*^2^ + *x* − 90 = 0]{.math.inline} [(*x*+10)(*x*−9) = 0]{.math.inline} [*x* + 10 − 10 = − 10 *x* − 9 + 9 = 0 + 9]{.math.inline} [*x* = − 10 *or* *x* = 9]{.math.inline} To Check: [*x*(*x*+1) = 90 *x*(*x*+1) = 90]{.math.inline} -10(-10+1) = 90 9(9+1) = 90 -10(-9) = 90 ✓ 9(10) = 90 ✓ **[Rational Algebraic Expressions]** **Rational Number** -- is any number that can be written as a ratio:[\$\\frac{a}{b}\$]{.math.inline}, where a and b are integers and [*b* ≠ 0.]{.math.inline} Examples: -98, 6, 325, -4.1, 0, and [\$\\frac{5}{6},\\frac{- 3}{5}, - \\frac{7}{8},\\frac{14}{- 3}\$]{.math.inline} **Algebraic Expression** -- a number, a variable, a sum, difference, or product that contains one or more variable. Examples: [*y*^2^ − 2*y* + 6, 2*c*^2^*d*, *x*, − 3]{.math.inline} **Rational Algebraic Expression** -- Is a ratio of two polynomials provided that the denominators is not equal to zero. In symbols;[\$\\frac{P}{Q}\$]{.math.inline} where P and Q are polynomials and[ *Q* ≠ 0]{.math.inline} **An algebraic Expression is not a Polynomial if:** 1\. The exponent of the variable is not a whole number (0, 1, 2, 3...). 2\. The variable is inside the radical sign. 3\. The variable is in the denominator. **Undefined Rational Expression** \- a rational expression is undefined when the denominator is equal to zero. \- the numerator being equal to zero is okay (the rational expression simply equals to zero). **Example:** [\$\\frac{{9x}\^{3} + 4x}{15x + 45}\$]{.math.inline} [15*x* + 45 = 0]{.math.inline} [15*x* = − 45]{.math.inline} [\$\\frac{15x}{15} = \\frac{- 45}{15}\$]{.math.inline} [*x* ≠ − 3]{.math.inline} **Simplifying Rational Algebraic Expression** \- means writing the lowest terms or simplest form. **To simplify rational algebraic expressions:** 1\) Factor the numerator completely. 2\) Factor the denominator completely. 3\) Simplify (reduce) any like factors (not terms) **Examples:** [\$1)\\text{\\ \\ }\\frac{{2x}\^{3}}{14x}\$]{.math.inline} = [\$\\frac{2 \\bullet x \\bullet x \\bullet x}{2 \\bullet 7 \\bullet x}\$]{.math.inline} = [\$\\frac{x\^{2}}{7}\$]{.math.inline} ; [*x* ≠ 0]{.math.inline} 2\) [\$\\ \\frac{7x + 35}{x\^{2} + 5x} = \\frac{7(x + 5)}{x(x + 5)} = \\frac{7}{x}\\ \\ ;x \\neq 0,\\ - 5\$]{.math.inline} 3\) [\$\\frac{x\^{2} + 3x - 4}{x\^{2} - x - 20} = \\frac{\\left( x + 4 \\right)\\left( x - 1 \\right)}{\\left( x + 4 \\right)\\left( x - 5 \\right)} = \\frac{x - 1}{x - 5}\\ ;x \\neq 5\$]{.math.inline} 4\) [\$\\frac{x\^{2} - 25}{25 - x\^{2}} = \\frac{x\^{2} - 25}{- 1( - 25 + x\^{2})} = - 1\$]{.math.inline} **[Operations on Rational Algebraic Expressions ]** **Multiplying Rational Algebraic Expressions** **To Multiply:** 1\. Factor the numerator and denominator of each fraction. 2\. Multiply the numerator and denominator of each fraction. 3\. Divide it the common factors. 4\. Write the answer in the simplest form. **Example:** [\$\\frac{x\^{2} + 3X}{x\^{2} - 2x - 3} \\bullet \\frac{x\^{2} - x - 2}{x\^{2} + 2x - 3}\$]{.math.inline} [\$= \\ \\frac{x\\left( x + 3 \\right) \\bullet \\left( x + 1 \\right)\\left( x - 2 \\right)}{\\left( x + 1 \\right)\\left( x - 3 \\right) \\bullet \\left( x + 3 \\right)\\left( x - 1 \\right)}\$]{.math.inline} [\$= \\ \\frac{x\^{2} - 2x}{x\^{2} - 4x + 3}\$]{.math.inline} **Dividing Rational Algebraic Expressions** **To Divide:** 1\. Multiply the divided by the reciprocal of the divisor. 2\. Multiply the numerators. Then multiply the denominators. 3\. Divide by the common factors. 4\. Write the answer in simplest from. **Example:** [\$\\frac{x - x\^{2}y}{z} \\div \\frac{2x - 2x\^{2}y}{z\^{2}}\$]{.math.inline} [\$= \\ \\frac{x - x\^{2}y}{z} \\bullet \\frac{z\^{2}}{2x - 2x\^{2}y}\$]{.math.inline} [\$= \\ \\frac{x\\left( 1 - xy \\right) \\bullet z \\bullet z}{z \\bullet 2x\\left( 1 - xy \\right)}\$]{.math.inline} [\$= \\frac{z}{2}\$]{.math.inline} **Adding & Subtracting Similar Rational Algebraic Expression** **To add/subtract similar rational expressions:** 1\. Add/Subtract the numerators and copy the denominator. 2\. Combine the like terms in the numerator. 3\. Factor the numerator and denominator if possible. 4\. Divide out common factors between the numerator and the denominator. 5\. Simplify the remaining expressions in the numerator and denominator. **Examples:** [\$\\frac{5}{2b} + \\frac{3}{2b}\$]{.math.inline} [\$= \\frac{5 + 3}{2b}\$]{.math.inline} [\$= \\ \\frac{8}{2b}\$]{.math.inline} [\$= \\frac{2 \\bullet 2 \\bullet 2}{2 \\bullet b}\$]{.math.inline} [\$= \\ \\frac{4}{b}\$]{.math.inline} [\$\\frac{9m\^{2}}{m - 2} - \\frac{m\^{2} + 16m}{m - 2}\$]{.math.inline} [\$= \\ \\frac{9m\^{2} - \\left( m\^{2} + 16m \\right)}{m - 2}\$]{.math.inline} [\$= \\ \\frac{9m\^{2} - m\^{2} - 16m}{m - 2}\$]{.math.inline} [\$= \\ \\frac{8m\^{2} - 16m}{m - 2}\$]{.math.inline} [\$= \\ \\frac{8m(m - 2)}{m - 2}\$]{.math.inline} [ = 8*m*]{.math.inline} **Adding and Subtracting Dissimilar Rational Algebraic Expressions** **To add/subtract dissimilar rational expressions:** 1\. Find the LCD. 2\. Use the LCD to make the rational algebraic expressions similar. 3\. Perform the steps in adding and subtracting similar rational algebraic expressions. **Examples:** [\$\\frac{3x}{x\^{2} - 2x + 1} + \\frac{4}{2x - 2}\$]{.math.inline} [*x*^2^ − 2*x* + 1 = (*x*−1)(*x*−1)]{.math.inline} [2*x* − 2 = 2(*x*−1)]{.math.inline} LCD: [2(*x*−1)(*x*−1)]{.math.inline} [\$= \\frac{2\\left( 3x \\right)}{2\\left( x - 1 \\right)\\left( x - 1 \\right)} + \\frac{4\\left( x - 1 \\right)}{2\\left( x - 1 \\right)\\left( x - 1 \\right)}\$]{.math.inline} [\$= \\ \\frac{6x + 4x - 4}{2\\left( x - 1 \\right)\\left( x - 1 \\right)}\$]{.math.inline} [\$= \\frac{10x - 4}{2\\left( x - 1 \\right)\\left( x - 1 \\right)}\$]{.math.inline} [\$= \\ \\frac{2(5x - 2)}{2(x - 1)(x - 1)}\$]{.math.inline} [\$= \\ \\frac{5x - 2}{x\^{2} - 2x + 1}\$]{.math.inline} **Solving Problems Involving Rational Algebraic Expressions** **Steps:** 1\. Read and understanding the problem. Identify what is given and what is being unknown. Choose a variable to represent the unknown number. 2\. Express the other unknowns, if there are any, in terms of the variable chosen in step 1. 3\. Whate an equation to represent the relationship among the given and the unknowns. 4\. Solve the equation for the unknown and use the solution to find the quantities being asked being asked. 5\. Check. **Examples:** **A. Number Problem** If the same number is added to both numerator and denominator of the fraction[\$\\frac{1}{2}\$]{.math.inline}, the result is[\$\\ \\frac{3}{4}\\text{.\\ }\$]{.math.inline}Find the number. **Step 1**: Let x = the number **Step 2:** 1+x = numerator 2+x = denominator **Step 3:** [\$\\frac{1 + x}{2 + x} = \\frac{3}{4}\$]{.math.inline} **Step 4:** [\$\\frac{1 + x}{2 + x} = \\frac{3}{4}\\ \$]{.math.inline} Equation [4(1+*x*) = 3(2 + *x*)]{.math.inline} Cross Multiply [4 + 4*x* = 6 + 3*x*]{.math.inline} Distributive Property [4 + (−4) + 4*x*(−3)= 6 + (−4) + 3*x* + ( − 3*x*)]{.math.inline} Addition Property of equality [*x* = 2]{.math.inline} Combine like terms **B. Age Problem** \- are algebraic problems that deal with the ages of people currently, in the past, or in the future. Five years ago, John's age was half of the age will be in 8 years. How old is he now? **Step 1:** Let x be John's age **Step 2**: x-5 is John's age five years ago [\$\\frac{1}{2}(x + 8\$]{.math.inline} is half of the age he will be in 8 years **Step 3:** [\$x - 5 = \\frac{1}{2}(x + 8)\$]{.math.inline} Step 4: [\$x - 5 = \\frac{1}{2}(x - 8)\$]{.math.inline} Equation [2(*x*−5) = *x* + 8]{.math.inline} Cross Multiply [2*x* − 10 = *x* + 8]{.math.inline} Distributive Property [2*x* + (−*x*) − 10 + 10 = *x* + (−*x*) + 8 + 10]{.math.inline} Addition Property of Equality [*x* = 18]{.math.inline} Combine Like terms **C. Work Problem** \- the formula for work problem that has two persons is: [\$\\frac{1}{t\_{1}} + \\frac{1}{t\_{2}} = \\frac{1}{t\_{3}}\$]{.math.inline} t~1~ -- is the time spent by the first person t~2~ -- is the time spent by the second person t~3~ -- is the time spent by both Joey can mow the lawn in 40 minutes and Pete can mow the lawn in 60 minutes. How long will it take them to mow the lawn together? Step 1: Let x be time to mow the lawn together Step 2: [\$\\frac{1}{40}\$]{.math.inline} time taken by Joey to mow the lawn alone [\$\\frac{1}{60}\$]{.math.inline} time taken by Pete to mow the lawn alone [\$\\frac{1}{x}\$]{.math.inline} time taken by Joey and Pete to mow the lawn together Step 3: [\$\\frac{1}{40} + \\frac{1}{60} = \\frac{1}{x}\$]{.math.inline} Step 4: [\$\\frac{1}{40} + \\frac{1}{60} = \\frac{1}{x}\$]{.math.inline} Equation [\$120x\\left( \\frac{1}{40} + \\frac{1}{60} \\right) = 120x(\\frac{1}{x})\$]{.math.inline} Multiply Bothe sides by 120x, the LCD of 40, 60, and x. [\$\\frac{120x}{40} + \\frac{120x}{60} = \\frac{120x}{x}\$]{.math.inline} Distributive Property [3*x* + 2*x* = 120*x*]{.math.inline} Simplify [5*x* = 120]{.math.inline} Combine like terms [\$\\frac{5x}{5} = \\frac{120}{5}\$]{.math.inline} Divide both sides by 5 [*x* = 24]{.math.inline} Simplify D. Speed/Travel Problem \- an object is said to be in uniform motion when it moves without changing its speed or rate. Formula: d = rt r = d/t t = d/r Macky won a two-day bicycle race. He travelled 60km each day and his average speed on the second day was doubled that of the first day. If Macky rode for a total of 6 hours, what was his average speed each day? Step 1: Let x be the speed of the first day. Step 2: Distance Speed(rate) time ------- ---------- ------------- ------------------------------------- Day 1 60 x [\$\\frac{60}{x}\$]{.math.inline} Day 2 60 2x [\$\\frac{60}{2x}\$]{.math.inline} Total 6 hours Step 3: [\$\\frac{60}{x} + \\frac{60}{2x} = 6\$]{.math.inline} Step 4: [\$\\left( \\frac{60}{x} + \\frac{60}{2x} = 6 \\right)2x\$]{.math.inline} Multiply both sides by the LCD of x and 2x. [120 + 60 = 12*x*]{.math.inline} Combine like terms [\$\\frac{180}{12} = \\frac{12x}{12}\$]{.math.inline} Divide by 12 [15 = *x*]{.math.inline} Simplify **[Rectangular Coordinate System]** **Rectangular coordinate system** \- it is defined by two perpendicular number lines that meet at the point of origin (0,0) and divide the plane into four regions called quadrants. \- the horizontal number line is called the x-axis and the vertical number line is called the y-axis. \- the Cartesian plane extends infinitely. \- each point in the coordinate system is defined by an ordered pair of the form (x,y) where x and y are real numbers. \- this pair of numbers is called coordinates of a point. \- the coordinates of a point determine the location of a point in the Cartesian coordinate plane by indicating its distances from the axes. \- the distance from the y-axis is called the abscissa or the x- coordinate and that is the first number in an ordered pair, while the distance from the x-axis is called the ordinate or the y- coordinate and the second number in an ordered pair. Coordinate System - Quadrants, Sign Convention, Application & Types **Plotting of points:** To plot a point is to locate the position of coordinates in the Cartesian plane. Follow these steps in plotting points: a\. Locate the x-coordinate along the x-axis. Draw an imaginary line parallel to the y-axis. b\. Locate the y-coordinate along the y-axis. Draw an imaginary line parallel to the x-axis. c\. Mark the intersection of the two imaginary lines with a dot. d\. Label the dot or point using a capital letter

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