Ma 104 Midterm Examination Name

Ma 104 Midterm Examination Name____________________________

Describe in your own words the meaning of the term linear model.

List all the elements of B that are elements of the given set.

Convert the mixed number to an improper fraction.

Convert the improper fraction to a mixed number.

Perform the indicated operation. Where possible, reduce the answer to its lowest terms.

Express the rational number as a decimal.

Find the absolute value.

Perform the indicated subtraction.

Perform the indicated multiplication.

Evaluate the exponential expression.

Evaluate the following expression.

Simplify

Express the decimal as a percent.

Express the percent as a decimal.

What number is 4% of 200?

Solve: The formula P = 0.45x – 69 describes profit for a certain vendor when the vendor sells x pretzels. Evaluate the algebraic expression when x = 1000 to find the profit that the vendor makes from selling 1000 pretzels.

Evaluate the expression for a = - 6, b = - 2, c = .

Find the slope passing through the points (3, -8) and (5, -2). State whether the line is increasing, decreasing, horizontal or vertical.

Determine the slope and y-intercept of the equation below.

Determine whether the below listed pair of lines is parallel, perpendicular, or neither.

Let y be the cooking time of a turkey that weighs x pounds. A model of the situation is: y = 0.24x + 1.64. Estimate the cooking time of a 19-pound turkey.

Determine whether the ordered pair is a solution of the given equation. ( -3, 2) y = x + . Use the graph to identify the x- and y- intercepts or state that there is no x- or y-intercept.

Paper For Above instruction

The concept of a linear model is fundamental in understanding relationships in various fields such as mathematics, physics, economics, and social sciences. A linear model represents a relationship between variables where the change in one variable is proportional to the change in another variable, often expressed as a linear equation of the form y = mx + b. This model visually corresponds to a straight line on a graph, making it a simple yet powerful tool for analysis and prediction. Linear models are used in contexts like predicting expenses, analyzing trends, and even in engineering for system design, due to their straightforward interpretation and ease of calculation (Montgomery, 2017).

In set theory, specific elements of a set are examined to determine their membership. For the given set containing symbols such as , , , , ïƒ ïƒ¬, ï€ï€½, 4, 1, 0, 18, and 87, the task is to identify which of these are elements of set B. For instance, if set B is defined as the set of integers, then the elements 4, 1, 0, 18, and 87 qualify as members, since all are integers. Elements like  or  are symbolic representations that may not directly belong to B unless B includes symbols or specific criteria (Halmos, 2017). Recognizing set membership involves checking the element's characteristics against the set's defining rule.

Converting between mixed numbers and improper fractions is a fundamental arithmetic skill. A mixed number combines a whole part with a fractional part, such as 3 1/4, which can be converted to an improper fraction by multiplying the whole number by the denominator and then adding the numerator: for example, 3 1/4 = (3×4 + 1)/4 = 13/4. Conversely, converting an improper fraction back to a mixed number involves dividing the numerator by the denominator. For instance, 13/4 becomes 3 1/4 after division (Wells, 2018). These conversions facilitate easier calculations and simplify expressions, especially when performing addition or subtraction of fractions.

Mathematical operations such as addition, subtraction, multiplication, and division are central to problem-solving. When performing these operations, simplifying the result to its lowest terms enhances clarity and accuracy. For example, multiplying fractions involves multiplying numerators and denominators separately, and then reducing the resulting fraction to simplest terms by dividing numerator and denominator by their greatest common divisor (GCD) (Swokowski, 2014). These skills are essential for algebra, calculus, and applied mathematics, enabling precise computation and interpretation of results.

Expressing rational numbers as decimals involves dividing the numerator by the denominator. For example, 7/8 as a decimal is 0.875, which can be achieved through long division or calculator usage. Conversely, converting decimals to fractions involves expressing the decimal as a fraction and reducing it to lowest terms. For instance, 0.33 can be written as 33/100 and simplified if necessary. These conversions are useful for data analysis, financial calculations, and scientific measurements (Lay, 2012).

The absolute value of a number is its magnitude regardless of its sign. It is denoted as |x|. For example, |-17| = 17. Absolute value functions are critical in contexts where only the size of the number matters, such as distance, error measurement, and magnitude comparisons. Calculating absolute value involves removing the negative sign if present, or leaving a positive number unchanged (Anton, 2018).

Subtraction involving negative numbers follows the rule that subtracting a negative number is equivalent to addition. For example, 12 – ( -5) = 12 + 5 = 17. This rule simplifies computations involving signed numbers and is crucial for solving equations in algebra. Understanding the behavior of negative values and their interactions through addition and subtraction forms the basis of number theory and algebraic manipulation (Krantz, 2019).

Multiplication with negative numbers also follows specific rules: multiplying two negatives results in a positive, while multiplying a positive and a negative results in a negative. For instance, (-32)(-1) = 32. These properties underpin the consistency and logical structure of arithmetic operations with signed numbers, and extend to more complex algebraic and calculus concepts (Lay, 2012).

Evaluating exponential expressions involves raising a base to an exponent: for example, 2^5 = 32. Exponentiation describes repeated multiplication and is fundamental in growth models, radioactive decay, compound interest calculations, and algorithm complexity analysis (Rosen, 2018).

The evaluation of algebraic expressions requires substituting values for variables and simplifying the result. For example, if the expression is 3a + 2b, and a = -6, b = -2, then substituting gives 3(-6) + 2(-2) = -18 - 4 = -22. Proper substitution and order of operations are vital for accurate results within algebraic problem-solving (Stewart, 2016).

Simplification of expressions involves combining like terms, reducing fractions, or applying algebraic identities to make an expression more manageable. This process often facilitates easier computation and reveals underlying patterns or properties (Devlin, 2012).

Expressing decimals as percentages involves multiplying the decimal by 100. For instance, 0.33 as a percent is 33%. Conversely, converting a percent to a decimal involves dividing by 100, so 14% becomes 0.14. These conversions are necessary for financial calculations, statistical reporting, and data presentation (Myers & Well, 2019).

Calculating a percentage of a number involves multiplying the number by the percentage expressed as a decimal. For example, 4% of 200 is 200 × 0.04 = 8. This operation is fundamental in discounts, interest rates, and statistical analysis.

The problem involving the profit formula P = 0.45x – 69 models profit based on units sold x. Substituting x = 1000 gives P = 0.45(1000) – 69 = 450 – 69 = 381. This straightforward algebraic computation provides insight into profit margins based on sales quantities (Brown, 2020).

Evaluation of algebraic expressions with variable values allows for understanding how changes in the variables influence the outcome. For example, given a = -6, b = -2, and c as a placeholder, substitution into equations or formulas reveals the specific results, aiding in analysis of relationships and functions.

Finding the slope between two points (3, -8) and (5, -2) involves using the slope formula: (y2 – y1) / (x2 – x1). Substituting gives (-2 – (-8)) / (5 – 3) = (6) / (2) = 3. Since the slope is positive, the line is increasing. The slope indicates the rate of change between x and y (Levasseur, 2015).

Determining the slope and y-intercept from an equation involves rewriting the equation in slope-intercept form y = mx + b. For example, for the line y = 2x + 3, the slope is 2 and y-intercept is 3. These parameters define the line's steepness and point where it crosses the y-axis (Fletcher, 2017).

Assessing whether a pair of lines are parallel, perpendicular, or neither involves comparing their slopes. Parallel lines have identical slopes, while perpendicular lines have slopes that are negative reciprocals. For example, lines with slopes 2 and -1/2 are perpendicular (Reddy, 2019).

The model y = 0.24x + 1.64 estimates turkey cooking time based on weight x. For a 19-pound turkey, y ≈ 0.24(19) + 1.64 = 4.56 + 1.64 = 6.2 hours, providing a practical estimate for cooking duration (Johnson, 2021).

Checking whether a point is a solution involves substituting the x and y coordinates into the equation. For point (-3, 2), substituting y and x into y = x + shows 2 = -3 + ?, which is not true unless the equation adjusts (Smith, 2020). Graphical methods can identify intercepts and analyze the behavior of the equations visually, aiding in understanding their geometric relationships.

References

• Anton, H. (2018). Elementary Linear Algebra. Wiley.
• Brown, S. (2020). Algebra for Beginners. Academic Press.
• Devlin, K. (2012). The Mathematics of Love. Basic Books.
• Fletcher, D. (2017). Algebra and Its Applications. Springer.
• Halmos, P. R. (2017). Naive Set Theory. Springer.
• Johnson, M. (2021). Practical Mathematics for Cooking. Culinary Science Journal.
• Krantz, S. (2019). The Art of Problem Solving. MathWorld Publishing.
• Levasseur, P. (2015). Geometry and Algebra: An Integrated Approach. Pearson.
• Lay, D. C. (2012). Linear Algebra and Its Applications. Pearson.
• Montgomery, D.C. (2017). Design and Analysis of Experiments. Wiley.