Follow Order Of Operations To Simplify The Expression

Follow Order Of Operations To Simplify The Following Expression1 2

Follow order of operations to simplify the following expression: + 20 ´ 2 -10. Evaluate the expression using the values given for the variables: x = -3, and y = 5. Simplify the expression: b + b. Write your answer as a fraction in lowest terms. Use a proportion to set up and solve the following problem: If a 6 ft tall petrified stump casts a 34.2 ft long shadow, then how long is the shadow that a 9.2 ft tall adult elephant casts? Set up and solve an equation to answer the question below. Round your answer to the nearest tenth of a percent if necessary. 39 is what percent of 108? Find the percentage change. Round to the nearest tenth of a percent if necessary and state whether the change is an increase or a decrease. From $87.40 to $55, find the final price of the item below, given that the tax will be applied to the discounted price. Original price of a cell phone: $179.50, Discount: 30%, Tax: 6%. Use simple interest to find the ending balance, given the following values for principal, rate, and time: $58,000 at 9.7% for 8 years. Solve the equation: -2 + 4x = 34. Solve the equation: 72 = -3(8x - 1) + 3(5 + 2x). Solve the equation: 6x - 7 = 2x. Solve the inequality: -6.32 + 0.9v = v - 5.9. Solve the inequality: 10 p + 4 3(k - 6). Find the slope of the line assuming a scale of 1 unit per tick mark. Find the x- and y-intercepts of the line y = x - 3. Write the slope-intercept form of the line passing through points: (-2, -1) and (4, -5). Find at least three ordered pairs satisfying the equation and sketch the graph. Solve the system: -2 = x - 2 y using substitution. Write the point-slope form of the line through (5, -4), parallel to y = - 6/5 x - 1. Write the slope-intercept form of the line through (-5, -1), perpendicular to y = 5/4 x + 3. Solve the system by graphing: 3x - y = -4 and 4x + y = -3. Solve the system using elimination: 16x + 6 y = 4 and 8x - 2 y = 12. Kathryn and Shayna are selling pies; set up and solve the system to find the cost of each type of pie.

Paper For Above instruction

Introduction

Mathematics plays a crucial role in problem-solving across various fields, from simple arithmetic to complex equations. The set of problems provided covers several fundamental concepts, including order of operations, algebraic expressions, proportions, percentages, interest calculations, equations, inequalities, graphing, and systems of equations. Mastery of these topics is essential for developing analytical thinking skills and applying mathematical principles to real-world scenarios.

Order of Operations and Simplification

The initial problem involves applying the order of operations (PEMDAS/BODMAS) to simplify an expression. For example, the expression "+ 20 ´ 2 -10" needs to be carefully evaluated. The operations are multiplication first, then addition and subtraction, respecting their precedence. Assuming the expression is "20 ´ 2 - 10," multiplication is performed first: 20 ´ 2 = 40, then subtract: 40 - 10 = 30. This illustrates the importance of following the correct sequence to arrive at the correct answer.

Evaluating Expressions Using Variables

Evaluating algebraic expressions such as "x^2 + y + xy" requires substituting given variable values. With x = -3 and y = 5, substitute: (-3)^2 + 5 + (-3)(5). Calculating: 9 + 5 - 15 = -1. This showcases how substituting values simplifies the problem to basic arithmetic, emphasizing the importance of understanding variable manipulation.

Simplification of Algebraic Expressions and Fractions

Expressions like "b + b - 3" simplify to 2b - 3. When writing answers as fractions, such as simplifying complex fractions, it is crucial to reduce to lowest terms. For instance, if the fraction is 4/8, it simplifies to 1/2. Proper handling of fractions ensures clarity and correctness in mathematical communication.

Using Proportions in Real-World Contexts

A common application involves setting up proportions to solve problems related to similar figures or shadows. For example, given a shadow length of 34.2 ft for a 6 ft stump and a tall elephant, proportionality allows setting up: 6 / 34.2 = 9.2 / x. Cross-multiplying yields x = (9.2 × 34.2) / 6, which calculates the shadow length. Such problems demonstrate how ratios model real-world phenomena.

Percentages and Percent Change

Percent calculations, such as determining what percent 39 is of 108, involve dividing: (39 / 108) × 100 ≈ 36.1%. For percent change, the formula is ((new - old) / old) × 100. Transitioning from $87.40 to $55 shows a decrease: ((55 - 87.4)/87.4) × 100 ≈ -37.0%, indicating a 37% reduction. Accurate percentage calculations are vital in finance and data analysis.

Applying Discount and Tax

Computing the final price after discounts and taxes involves sequential calculations. For a $179.50 phone with a 30% discount, the discounted price is $179.50 × (1 - 0.30) = $125.65. Applying a 6% tax results in $125.65 × (1 + 0.06) ≈ $133.38. Understanding this sequence is essential for financial literacy.

Interest Calculations

Simple interest is calculated as I = P × r × t, where P is principal, r is rate, and t is time in years. For $58,000 at 9.7% for 8 years, interest = $58,000 × 0.097 × 8 = $45,088. The total amount after interest is $58,000 + $45,088 = $103,088, illustrating long-term investment growth.

Solving Equations and Inequalities

Linear equations such as -2 + 4x = 34 require isolating the variable: 4x = 36 → x = 9. Similarly, inequalities like 10 p + 4

Graphing and Analyzing Lines

Finding slopes involves calculating the change in y over the change in x between two points. For example, between points (-2, -1) and (4, -5), the slope m = (-5 + 1) / (4 + 2) = -4/6 = -2/3. Intercepts are determined by setting x or y to zero and solving. Writing equations in slope-intercept and point-slope forms enables graphing lines accurately.

System of Equations

Systems like -2 = x - 2 y can be solved via substitution or elimination. Substituting y from one equation into the other or adding equations to eliminate a variable results in solving for one variable, then back-substituting for the other. Graphing solutions visually confirms intersection points.

Application in Real-Life Scenarios

Real-world problems, such as calculating the cost of pies sold by Kathryn and Shayna based on their sales data, involve setting up systems of equations. Solving these yields the prices per pie type, linking algebra to everyday transactions. Such applications highlight the practical importance of these mathematical tools.

Conclusion

Overall, these problems encapsulate essential mathematics skills — from basic operations to advanced algebra — necessary for academic success and practical application. Developing proficiency in these areas enhances critical thinking, problem-solving ability, and quantitative literacy, vital for academic pursuits and everyday decision-making.

References

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