MATH 9 - Solving Linear Equations PDF

# MATH 9: Solving Linear Equations ## Page 1 **Name:** Key **Homeroom:** **Date:** **MATH 9** Solving Linear Equations ## Page 2 **Name:** **Homeroom:** **Date:** ## EXPRESSIONS - In mathematical statements, a variable is a letter or symbol to represent an unknown value. - An algebraic expre...

# MATH 9: Solving Linear Equations ## Page 1 **Name:** Key **Homeroom:** **Date:** **MATH 9** Solving Linear Equations ## Page 2 **Name:** **Homeroom:** **Date:** ## EXPRESSIONS - In mathematical statements, a variable is a letter or symbol to represent an unknown value. - An algebraic expression is a combination of letters, symbols and numbers to represent a mathematical operation. **Examples of algebraic expressions include:** - Double a number and add six : $2n+6$ - Five less than four times a value: $4v-5$ - An expression is different than an equation because an expression cannot be simplified until I know the value of the unknown. An equation can be solved. **Examples of algebraic equations include:** - $3n-12=-11$ - $4x=-100$ - $4a + 3b = 107$ - $17= 3x +2$. ## Page 3 **Name:** **Homeroom:** **Date:** ## A. Problems Involving Linear Equations - The cost to print flyers for a restaurant cost $250 as a flat fee, plus $1.25 per flyer. - Write an equation to represent this situation. - What is the cost of 2500 flyers? - How many flyers can be printed for $625? **Let $f$ be the number of flyers and $C$ be the cost.** - $250 + $1.25 f = C$ **For 2500 flyers:** - $250 + $1.25 (2500) = C$ - $250 + $3125 = C$ - $3375 = C$ **For $625:** - $250 + 1.25 f = 625$ - $1.25f = 375$ - $f = 300$ flyers - A cheese and tomato pizza costs $9.00. Each additional topping is $0.75. - Write an equation to represent this situation. - What is the cost for each pizzas with 1-5 toppings? - If a pizza cost $15.00, what might be the number of toppings? **Let $t$ be the number of toppings and $C$ be the cost.** - $9.00 + $0.75 t = C$ | **t** | **C** | |---|---| | 0 | $9.00 | | 1 | $9.75 | | 2 | $10.50 | | 3 | $11.25 | | 4 | $12.00 | | 5 | $12.75 | **For $15.00:** - $9.00 + 0.75 t = $15.00$ - $0.75t = 6.00$ - $t ≈ 8$ ## Page 4 **Name:** **Homeroom:** **Date:** ## B. EQUATIONS **Describe in mathematical terms: "Equality".** Equality means both sides of an equal sign are worth the same amount, no matter the steps or how many there are to. **Example:** - $1+2+3-4 = 2$ **Explain what it means to isolate the variable.** Isolating the variable means to perform a series of inverse operations to get an unknown amount on one side of the equal sign and the variable on the opposite side. **Example:** - $3n+ 5 = 26$ - $-5$ - $-5$ - $3n = 21$ - $n=7$ **Share your understanding of preserving equality.** In the above example, the same operations were made at the same time on both sides of the equal sign. This keeps the equation in balance. ## Page 5 **Name:** **Homeroom:** **Date:** ## C. Solving One-Step Equations: $x + a = b$ or $x - a = b$ **Solve using addition and/or subtraction.** | Equation | Solution | |---|---| | $x - 3 = 11$ | $x = 14$ | | $n - 7 = 8$ | $n = -1$ | | $3 = y - 10$ | $13 = y$ | | $m - 5 = -4$ | $m = +1$ | | $y + 2 = -7$ | $y = -9$ | | $x + 3 = 9$ | $x = 6$ | | $-8.8 = w - 1.1$ | $-7.7 = w$ | | $12 + q = 10$ | $q = -2$ | | $10.3 = x - 5.4$ | $15.7 = x$ | | $-3 = s + 5$ | $-8=s$ | | $4.6 = t - 1.4$ | $6.0 = t$ | | $x +1.5 = 3.5$ | $x = 2$ | **Solve using multiplication and/or division.** $ax=b$, $\dfrac{x}{a} = b$, $\frac{a}{x}= b$ | Equation | Solution | |---|---| | $6x = 12$ | $x = 2$ | | $7z = -14$ | $z = -2$ | | $\dfrac{y}{7} = 4$ | $y = 28$ | | $\dfrac{x}{3} = 9$ | $x = 27$ | | $-6t = -24$ | $t = + 4$ | | $3m = -6.3$ | $m = -2.1$ | | $\dfrac{4}{w} = 1.5$ | $6 = w$ | | $-8.4 = 0.2x$ | $-42 = x$ | | $\dfrac{x}{5}= 2.4$ | $x = 12$ | ## Page 6 **Name:** **Homeroom:** **Date:** ## Translate each phrase into an algebraic expression or equation. | Phrase | Expression/Equation | |---|---| | An unknown number tripled | $3n$ | | Thirty-six is added to a number | $36 + n$ | | A person's age increased by four is twenty | $a + 4 = 20$ | | The sum of a number and four is sixteen | $n + 4= 16$ | | One-sixth of a number is ten | $\dfrac{n}{6} =10$ | | Six less than twice a number | $2n - 6$ | | Forty is five more than double a number | $40 = 5 + 2n$ | | The product of six and a number decreased by five | $6n - 5$ | | The sum of three consecutive numbers | $n + (n + 1) + (n + 2) = 3n + 3$ | | The sum of three consecutive odd numbers | $n + (n + 2) + (n + 4)$ **where n is odd** | **Write an algebraic expression** - If thirty is the first number, write the next three integers. - $n+1$ - $n+2$ - $n+3$ - If 43 is the first number, write the next two consecutive odd numbers. - $n + 2$ - $n + 4$ - If $x$ is an even integer, write the next two consecutive even integers. - $x + 2$ - $x + 4$ - A board is cut into two pieces and one piece is three cm longer than the other. Express their lengths algebraically. - **1st piece:** $n$ - **2nd piece:** $n + 3$ - The length of a rectangle is 5 meters more than twice the width. What is the width of the rectangle? - **w** and what is the length of the rectangle? - **2w + 5** - A boy has fourteen dimes and nickels. How many of each has he? - **dimes** - **nickels** **Solve the following by writing the equation and then solve.** - When a number is multiplied by five and the product increased by thirteen, the result is seventy-three. What is the number? - $5n + 13 = 73$ - $n = 12$ - Eric earned $\dfrac{2}{5}$ of the profits of the canteen on the weekend. His earnings were $620. What was the total profit earned in the canteen? - $\dfrac{2}{5}P = $620$ - $P = $1550$ ## Page 7 **Name:** **Homeroom:** **Date:** ## D. Solving Two-Step Equations: $(Ax + B = C)$: Solve the following equations: | Equation | Solution | |---|---| | $2x + 4 = 10$ | $x = 3$ | | $3x + 1 = 7$ | $x = 2$ | | $\dfrac{y}{3} + 1 = 6$ | $y = 15$ | | $\dfrac{y}{2} - 3 = 7$ | $y = 20$ | | $4x - 3 = 9$ | $x = 3$ | | $\dfrac{m}{4} - 1 = -3$ | $m = -8$ | | $\dfrac{x}{3} - 2 = 2$ | $x = 12$ | | $6x + 1 = 7$ | $x = 1$ | | $\dfrac{m}{2} + 3 = -1$ | $m = -8$ | | $-2x + 2 = -2$ | $x = -2$ | ## Page 8 **Name:** **Homeroom:** **Date:** ## Problem solving: 2 step equations 1. The cost of a pizza is $8.50, plus $1.35 per topping. How many toppings are on a pizza that costs $13.90? - $C = $8.50 + 1.35t$ - $t = 4$ toppings 2. Bob paid $34.95 to rent a car for the day, plus $0.12 for each kilometer he drove. The total rental cost before tax was $55.11. How many kilometers did Bob drive? - $C = $34.95 + $0.12k$ - $k = 168$ km 3. On Saturday morning, Mark had a quarter of his weekly allowance left. He spent a total of $6.50 on bus fare and a freshly squeezed orange juice on Saturday. He then had $2.25 left. How much is his weekly allowance? - $\dfrac{a}{4} = 6.50 + 2.25$ - $a = $35.00$ 4. Nadia had a summer job in an electronics store. She paid $400 per week, plus 55% commission on the total value of her sales. - One week, the store was not busy, Nadia only earned $510.30. What was the total value of her sales that week? - $P = $400 + 0.55(s)$ - $s = $200.55$ - Nadia's average earnings are $780 per week. What is the average value of her weekly sales? - $P = $400 + 0.55(s)$ - $s = $690.91$ 5. Winnipeg had 209 days of precipitation. That is 2.2 times the number of days that Calgary had precipitation. How many days did Calgary have precipitation? - $209 = 2.2d$ - $d = 95$ days 6. The bill for 6 hockey tickets was $390.00 before taxes. This included a handling charge of $5.00 for each ticket. What was the base price for each ticket with no taxes or handling fee? - $390 = 6(t + 5)$ - $t = $60.00$ ## Page 9 **Name:** **Homeroom:** **Date:** ## E. Solve Equations with Variables on Both Sides: $(Ax = B + Cx$ and $Ax + B = Cx + D)$ Solve the following equations: | Equation | Solution | |---|---| | $4x - 6 = 2x - 18$ | $x = - 6$ | | $3x - 2 = 6x - 10$ | $x = \dfrac{8}{3}$ | | $2 -3n = 2n + 7$ | $n = -1$ | | $13 - 3x = 4 - 2x$ | $x = 9$ | | $-2.5k - 2 = -5.7k + 6$ | $k=2.5$ | | $6.4 - 9.3b = 25.3 - 3.9b$ | $b=-3.5$ | | $3x - 8 = 7x + 4$ | $x = -3$ | | $8x - 9 = 25 - 9x$ | $x = 2$ | **Simplify first then isolate the variable by doing its opposites to both sides.** | Equation | Solution | |---|---| | $x + x + 2x + 7 + 9 = 36$ | $x = 5$ | | $4x + 4.2 + 2x = 9.7 + x$ | $x = 1.1$ | | $6y = 2y + y + 15$ | $y = 5$ | | $3x + x - 2.4 = x - 6.8 + 1.2$ | $x = -1.06$ | ## Page 10 **Name:** **Homeroom:** **Date:** ## G. Solve Equations using the distributive property: $a(x + b) = c$ OR $a(bx + c) = d(ex +f)$ Solve the following equations using the distributive property: | Equation | Solution | |---|---| | $2(x - 5) = 12$ | $x = 11$ | | $3(x + 6) = 42$ | $x = 8$ | | $4(x - 2) = 32$ | $x = 10$ | | $4(x + 3) = 2(x + 8)$ | $x = 2$ | | $5(x + 3) = 2(x + 9)$ | $x = 1$ | | $6(x - 4) = 2(x + 8)$ | $x = 10$ | | $6(x - 7) = 54$ | $x = 16$ | | $8(x + 3) = 64$ | $x = 5$ | | $9(x + 2) = 36$ | $x = 2$ | | $7(x - 4) = 5(x - 2)$ | $x = 9$ | | $2(x + 3) = 5(x - 6)$ | $x = 12$ | | $3(7 - x) = 5(x + 7)$ | $x = -1.75$ | ## Page 11 **Name:** **Homeroom:** **Date:** ## Create and solve an equation for each problem! 1. Video club members pay an annual membership fee of $40.00, which allows them to rent movies for $3.00 instead of $5.00. What is the least number of videos that a club member must rent during a year to save money from the membership? - $40 + 3m = 5m$ - $m = 20$ movies 2. A large pizza with tomato sauce and cheese costs $8.50, plus $1.25 for each additional topping. A customer orders a large pizza and is charged $14.75. How many toppings did the customer order? - $8.50 + 1.25t = 14.75$ - $t = 5$ toppings 3. Vianne took 3 bottles of water and 5 bottles of juice to a family picnic. Each bottle of juice contained 0.5 L. The total volume of water and juice was 4.75 L. What was the volume of 1 bottle of water? - $3w + 5(0.5) = 4.75$ - $w = 0.75$ L 4. During a ski jump, Maiko is airborne for 9.4 s. If her speed during the jump is 38.8 m/s, how far did Maiko jump, to the nearest hundredth of a metre? - $d = (38.8)(9.4)$ - $d = 364.72$ m 5. Winnipeg had 209 days of precipitation. That is 2.2 times the number of days that Calgary had precipitation. How many days did Calgary have precipitation? - $209 = 2.2d$ - $d = 95$ days 6. The bill for 6 hockey tickets was $390.00 before taxes. This included a handling charge of $5.00 for each ticket. What was the base price for each ticket with no taxes or handling fee? - $390 = 6(t + 5)$ - $t = 60$ ## Page 12 **Name:** **Homeroom:** **Date:** 7. The expression $180(n - 2)$ represents the sum of the interior angles in a polygon with n sides. Suppose the sum of its interior angles is 720°. How many sides does the polygon have? - $180(n - 2) = 720$ - $n = 6$ sides 8. At the end of the week, Francesca had a quarter of her babysitting money left after spending $16.05 on a shirt and another $1.95 on chips. How much did she earn babysitting? - $\dfrac{1}{4}b = 16.05 + 1.95$ - $b = $72.00$ 9. For a fit and healthy person, the maximum safe heart rate during exercise is approximately related to their age by the formula $r = \dfrac{4}{5}(220 - a)$. In this formula, $r$ is the maximum safe heart rate in beats per minute, and $a$ is the age in years. At what age is the maximum safe heart rate 164 beats/min? - $164 = \dfrac{4}{5}(220 - a)$ - $a = 15$ years old 10. Two rental halls are considered for a wedding. Hall A costs 60 per person. Hall B costs $3000, plus $40 per person. Determine the number of people for which the halls will cost the same to rent. - $60p = 3000 + 40p$ - $p = 150$ people 11. A taxi charges $4.25 plus $1.77 per kilometer traveled. If the total fare costs $37.88. Determine the number of kilometers the taxi traveled? - $4.25 + 1.77k = 37.88$ - $k = 19$ km 12. A rectangle has a width of $s + 1$ and a length of $2s + 5$. The perimeter of the rectangle is equal to 40.5 cm. What is the length of each side of the rectangle? - $P = 2(l + w)$ - $w = 5.75$ cm - $l = 14.5$ cm