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Problem Solving with linear models

Problem Solving with Linear Models" refers to the use of linear equations and relationships to address various real-world challenges and find solutions.

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Problem Solving with linear models

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  1. 0 0 PROBLEM SOLVING WITH LINEAR MODELS Solving word problems involving linear graphs 0 0 WWW.SUDO24.COM

  2. LESSON OUTLINE Linear Models and Equations 0 Interpreting Linear Models WWW.SUDO24.COM

  3. 0 CAN ALL DATA SETS BE SHOWN ON A GRAPH IN AN “ORDERLY” MANNER? WWW.SUDO24.COM

  4. 0 LEARNING OUTCOMES Intercept: the point where the line intersects the y-axis Identify and interpret the intercept of a linear model. Slope: the ratio of change in y to change in x between two points in a line Calculate and interpret the slope of a linear model. State the equation of the line in slope-intercept form. WWW.SUDO24.COM

  5. NOT ALL DATA SETS CAN LEAD TO AN “ORDERED” SET. A set of data may be scattered as the one shown in the graph but can be represented by a line of best fit. LINE OF BEST FIT: A straight line that best represents a set of scattered data. WWW.SUDO24.COM

  6. LINE OF BEST FIT 50 TEMPERATURE In an experiment for your Science class, you investigated how fast a given liquid changes its temperature. 40 30 (°C) 20 After gathering the data, your group drew the line of best fit as shown. 10 0 0 2 4 6 8 10 12 14 16 0 0 TIME (min) Can you identify the intercept? WWW.SUDO24.COM 0

  7. IDENTIFYING THE INTERCEPT OF THE LINE 50 TEMPERATURE 40 30 Recall that the y-intercept refers to the point where the line intersects the y-ax is. (°C) 20 10 Looking at the graph, the line intersects the y-axis at 25. 0 2 4 6 8 10 12 14 16 TIME (min) The y-intercept is thus 25. WWW.SUDO24.COM

  8. INTERPRETING THE INTERCEPT OF THE LINE 50 TEMPERATURE 40 What information can the y-intercept tell us? 30 (°C) 20 The y-intercept is also the value of y when x = 0. 10 0 2 4 6 8 10 12 14 16 Looking at the graph again, we can then say that the temperature of the liquid before heating (time = 0), is 25°C. TIME (min) WWW.SUDO24.COM

  9. CALCULATING THE SLOPE OF THE LINE TRY TO SOLVE FOR THE SLOPE OF THE LINE! Slope is the ratio of the change in y and the change in x. 50 TEMPERATURE change in y change in x 40 (degrees Celsius) slope (m) = 30 20 This can be calculated using two poi nts on the line with the formula: 10 0 2 4 6 8 10 12 14 16 y - y x - x 2 2 1 slope (m) = TIME (min) 1 WWW.SUDO24.COM

  10. CALCULATING THE SLOPE OF THE LINE y - y x - x ANSWER KEY 2 1 slope (m) = 50 2 1 TEMPERATURE 40 (degrees Celsius) Using the marked two points on the line: 30 40 - 35 7 - 5 20 slope (m) = 10 m = 2.5 0 2 4 6 8 10 12 14 16 TIME (min) This slope tells us that the temperature increases by 2.5°C every minute WWW.SUDO24.COM

  11. EQUATION OF THE LINE IN SLOPE-INTERCEPT FORM 0 0 Equation of a line in slope intercept form: y = mx+ b Using the y-intercept (b) and the slope (m), the line of best fit may be represented through an equation. 0 WWW.SUDO24.COM

  12. STATING THE EQUATION OF THE LINE 50 TEMPERATURE 40 Given these: b = 25 (degrees Celsius) m = 2.5 30 20 The equation of the line is: y = mx+ b 10 0 2 4 6 8 10 12 14 16 y = 2.5x+ 25 TIME (min) This equation can now be used to predict the temperature at a given time. WWW.SUDO24.COM

  13. Asked: Temperature (y) at 15 minutes (x) Solution: using the equation of the line ANSWER KEY ACTIVITY: USING THE EQUATION OF THE LINE y = 2.5x+ 25 y = 2.5 (15) + 25 y = 62.5°C Predict the temperature of the liquid after 15 minutes of heating. y = 2.5x+ 25 After 15 minutes of heating, the temperature of the liquid is predicted to be 62.5°C. WWW.SUDO24.COM

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