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Understanding Equations with Two Variables: Solutions and Graphing Techniques

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In this lesson, we explore how to find solutions for equations with two variables, such as ( y = 3x + 4 ). An example demonstrates finding a solution by substituting specific values into the equation. We discuss the significance of graphing these equations, highlighting that they can represent many solutions on a coordinate plane. The lesson includes examples of linear equations, showcasing how to create tables of values for graphing. Understanding the fundamental concepts of equations and their graphical representations is essential for mastering algebra.

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Understanding Equations with Two Variables: Solutions and Graphing Techniques

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  1. Section 8.2 Equations with Two Variablespages 405 - 409

  2. In today’s lesson, we will find solutions of equations with two variables, such as y = 3x + 4. An _______ ____ that makes such an equation a true statement is a ________of the equation.

  3. Example 1: Finding a Solution Find the solution of y = 3x + 4 for x = -1. y = 3x + 4 Replace x with -1. y = 3(-1) + 4 Multiply. y = -3 + 4 Add. y = 1 So, a solution of the equation is (-1,1)

  4. Why does the last line of the example say, “A solution of the equation…” instead of “The solution of the equation…?”

  5. Example 2 The equation t = 21 - 0.01nmodels the normal low July temperature in degrees Celsius at Mount Rushmore, South Dakota. In the equation, t is the temperature at n meters above the base of the mountain. Find the normal low July temperature at 700m above the base.

  6. Graphing Equations with Two Variables An equation with two variables can have many solutions. One way to show these solutions is to graph them, which also gives the graph of the equation. A ______ ________ is any equation whose graph is a line. The coordinates of every point on a line in a coordinate plane make the equation of the line a true statement.

  7. Graphing a Linear Equation Graph y = 4x – 2. • Make a table of values to show ordered pair solutions.

  8. y 5 y (-2,-10) • Graph the ordered pairs. Draw a line through the points. 8 6 (0,-2) 4 x (2,6) 2 -8 -6 -4 -2 2 4 6 8 -2 -4 -6 -8

  9. If you use the vertical-line test on the previous graph, you see that every x-value has exactly one y-value. This means that the relation y = 4x – 2 is a ________. A linear equation is a ________ unless its graph is a vertical line.

  10. Graphing a Linear Equation Graph y = 2x + 1. • Make a table of values to show ordered pair solutions.

  11. Graph the ordered pairs. Draw a line through the points. y 5 y 8 (-2,-3) 6 4 x 2 (0,1) -8 -6 -4 -2 2 4 6 8 -2 (2,5) -4 -6 -8

  12. Graphing a Linear Equation Graph y = 1/2 x + 4. • Make a table of values to show ordered pair solutions.

  13. Graph the ordered pairs. Draw a line through the points. y 5 y 6 (-2,3) 5 4 3 (0,4) 2 x 1 (2,5) -4 -3 -2 -1 1 2 3 4 -1 -2

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