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Follow order of operations to simplify the following expression: + 20 × 2 - 10. Evaluate the expression using the values given for the variables. x² + y + xy; use x = -3, and y = 5. Simplify the expression. b + b - 3. Simplify the expression and write your answer as a fraction in lowest terms: + 12. Use a proportion to set up and solve the following problem. Round your answer to the nearest tenth if necessary: 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 given in #10: -2 + 4x = 34. Solve the equation given in #11: 72 = -3(8x - 1) + 3(5 + 2x). Solve the equation given in #12: 6x - 7 = 2x. Solve the equation given in #13: -6.32 + 0.9v = v - 5.9. Solve the inequality in #14: 10p + 4 3(k - 6). Find the slope of the line. Assume that the scale on each axis is 1 unit per tick mark. Find the x- and y-intercepts of the line represented by the equation below. Plot both intercepts on a coordinate system and draw the line through them: x - 3y = -3. Write the slope-intercept form of the equation of the line through the given points: through (-2, -1) and (4, -5). Find at least three ordered pairs that satisfy the equation and sketch the graph of the line through them. -2 = x - 2y. Solve the system of equations by the method of substitution: -4x + 3y = -11; y = 5x. Write the point-slope form of the equation of the line described: through (5, -4), parallel to y = - 6/5 x - 1. Write the slope-intercept form of the equation of the line described: through (-5, -1), perpendicular to y = 5/4 x + 3. Solve the system by graphing both lines on one coordinate system: 3x - y = -4; 4x + y = -3. Solve the system of equations by the method of elimination: 16x + 6y = 4; 8x - 2y = 0. Kathryn and Shayna are selling pies for a school fundraiser. Customers can buy apple pies and pumpkin pies. Kathryn sold 3 apple pies and 13 pumpkin pies for a total of $164. Shayna sold 9 apple pies and 5 pumpkin pies for a total of $118. Find the cost of each one apple pie and one pumpkin pie.

Paper For Above instruction

Algebraic operations and problem-solving are essential skills for understanding mathematical relationships and applying them to real-world scenarios. This paper explores various algebraic concepts, including order of operations, evaluation of expressions, proportions, percentages, simple interest, solving linear equations and inequalities, graphing lines, and solving systems of equations, illustrating their relevance through practical problems.

Order of Operations and Expression Evaluation

To simplify mathematical expressions correctly, applying the order of operations is vital. The expression 20 × 2 - 10 involves multiplication followed by subtraction, resulting in 40 - 10 = 30. When evaluating expressions with variables, such as x² + y + xy with x = -3 and y = 5, substitute the values to find: (-3)² + 5 + (-3)(5) = 9 + 5 - 15 = -1. Simplifying algebraic expressions like b + b - 3 results in 2b - 3, which can be expressed in lowest terms as a linear expression.

Proportions and Practical Applications

Using proportions helps solve real-life problems like shadow length calculations. For example, given a 6 ft tall tree casting a 34.2 ft shadow, and a 9.2 ft tall elephant casting an unknown shadow, set up the proportion: 6/34.2 = 9.2/x, leading to x = (9.2 * 34.2) / 6 ≈ 52.44 ft. In percentage calculations, determining what percent 39 is of 108 involves the ratio (39 / 108) × 100 ≈ 36.1%. Such calculations are fundamental for understanding ratios and proportions in various contexts.

Percent Change and Financial Calculations

Calculating percentage change between two values, such as a price decline from $87.40 to $55, involves the formula: ((Old Price - New Price) / Old Price) × 100 ≈ ((87.40 - 55) / 87.40) × 100 ≈ 37.02% decrease. When factoring sales and taxes, the final price is computed by applying discounts and then adding tax. For instance, a $179.50 phone with a 30% discount yields a price of $125.65, and adding 6% tax results in $133.18. These calculations demonstrate the significance of sequential percentage operations.

Interest and Solving Linear Equations

Using simple interest formulas, such as I = P × r × t, helps determine earnings over a period. For a principal of $58,000 at an annual rate of 9.7% over 8 years, the interest is: 58,000 × 0.097 × 8 ≈ $45,088. The total amount after interest is P + I ≈ $103,088. Solving linear equations like -2 + 4x = 34 involves isolating the variable: 4x = 36, thus x = 9. These fundamental skills enable problem-solving in various quantitative contexts.

Solving Linear Equations and Inequalities

Equations such as 72 = -3(8x - 1) + 3(5 + 2x) require expansion and simplification, leading to 72 = -24x + 3 + 48x + 15, which simplifies to 72 = 24x + 18, giving 24x = 54, and so x = 2.25. Inequalities like 10p + 4

Graphing Lines, Intercepts, and Equations

The slope of a line can be calculated using two points, for example, (-2, -1) and (4, -5). The slope is (y2 - y1) / (x2 - x1) = (-5 + 1) / (4 + 2) = (-4) / (6) = -2/3. The x- and y-intercepts are found by setting y=0 and x=0 respectively. For the equation x - 3y = -3, the x-intercept occurs at x = -3 when y=0, and the y-intercept at y=1 when x=0. Plotting these points and drawing a line demonstrates linear relationships visually.

Writing Equations and Solving Systems

Given two points, the line's equation in slope-intercept form is y = mx + b. For points (-2, -1) and (4, -5), slope m = -2/3, and substituting one point yields b = 1. The equation is y = -2/3 x + 1. Systems of equations, such as -4x + 3y = -11 and y = 5x, can be solved by substitution: substituting y = 5x into the first yields -4x + 3(5x) = -11, leading to x = -1, and then y = -5. The intersection point is (-1, -5). For equations like 3x - y = -4 and 4x + y = -3, addition eliminates y, allowing the solving of the system efficiently.

Application: Pie Sales Problem

The problem involving Kathryn and Shayna’s pie sales translates to a system of two equations: 3a + 13p = 164 and 9a + 5p = 118, where a is the price of an apple pie and p the pumpkin pie. Solving these simultaneously, substitute from the first equation or use elimination. Multiplying the first by 3 and the second by 1 to align coefficients, leads to 9a + 39p = 492 and 9a + 5p = 118. Subtracting the second from the first gives 34p = 374, so p = 11. This yields a = (164 - 13*11)/3 ≈ 19.33. The prices are approximately $19.33 per apple pie and $11 per pumpkin pie, respectively, illustrating how systems of equations model real-world transactions.

Conclusion

Mastery of algebraic operations, proportional reasoning, percentage calculations, and solving equations and inequalities equips students with the tools necessary to analyze and interpret quantitative information. These skills are not only fundamental in mathematical contexts but are also applicable in everyday decision-making, financial literacy, and scientific problem-solving. Applying these concepts through diverse problems reinforces understanding and demonstrates their practical relevance in various fields.

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