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Understanding Linear Systems: Standard Form vs. Slope-Intercept Form

This guide explores the two primary forms of linear equations: Slope-Intercept Form (y = mx + b) and Standard Form (Ax + By = C). It covers how to recognize and graph these equations using their intercepts. By finding x and y-intercepts, you can effectively graph systems of equations and identify intersections, which represent solutions to linear systems. This guide includes practical examples to help you understand the concepts and visualize the relationships between linear equations.

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Understanding Linear Systems: Standard Form vs. Slope-Intercept Form

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  1. Graph Linear Systems Written in Standard Form

  2. Types of Linear Equations • Slope Intercept Form: y = mx + b • You have used this one the most. • If you have your slope and y-intercept, you can graph a line or even a system of equations (two lines). • Standard Form: Ax + By = C • “A” is the coefficient of “x.” • “B” is the coefficient of “y.” • “C” is a number (a constant)

  3. What type of Equation is this? • y = 2x -9 • 3x – 4y = 18 • -x + 19y = 5 • y = ½ x + 4 • 14x + y = 3 • y = -2/3x – 9/2 • Slope-Intercept Form • Standard Form • Standard Form • Slope-Intercept Form • Standard Form • Slope-Intercept Form

  4. Standard Form: Ax + By = C • To graph an equation in standard form, you use the x- and y-intercepts. • The x-intercept is: “What is x if y is zero?”(# , 0) • The y-intercept is: “What is y if x is zero?”(0, #)

  5. Find the x- and y- intercepts of the following equations: • 4x + 2y = 12 • 3x – y = 6 • -5x + 4y = 20 • 9x – 12y = -36 • (3, 0) & (0, 6) • (2, 0) & (0, -6) • (-4, 0) & (0, 5) • (-4, 0) & (0, 3)

  6. It is where the two lines intersect. What does “Solving a Linear System” mean?

  7. Graph to solve the linear system. 2x – y = 2 4x + 3y = 24 • Since the equations are in standard form, find the x- and y-intercepts to graph. 2x – y = 2 2x – 0 = 2 2x = 2 x = 1 (1, 0) 2x – y = 2 2(0) – y = 2 -y = 2 y = -2 (0, -2) 4x + 3y = 24 4x + 3(0) = 24 4x = 24 x = 6 (6, 0) 4x + 3y = 24 4(0) + 3y = 24 3y = 24 y = 8 (0, 8)

  8. Graph to solve the linear system. 2x – y = 2 Intercepts are (1, 0) & (0, -2) 4x + 3y = 24 Intercepts are (6, 0) & (0, 8) (3, 4) is the solution to this system of linear equations. Where do the lines intersect?

  9. Graph to solve the linear system. -4x – 2y = -12 4x + 8y = -24 • Since the equations are in standard form, find the x- and y-intercepts to graph. -4x – 2y = -12 -4x – 2(0) = -12 -4x = -12 x = 3 (3, 0) -4x – 2y = -12 -4(0) – 2y = -12 -2y = -12 y = 6 (0, 6) 4x + 8y = -24 4x + 8(0) = -24 4x = -24 x = -6 (-6, 0) 4x + 8y = -24 4(0) + 8y = -24 8y = -24 y = -3 (0, -3)

  10. Graph to solve the linear system. -4x – 2y = -12 Intercepts are (3, 0) & (0, 6) 4x + 8y = -24 Intercepts are (-6, 0) & (0, -3) (6, -6) is the solution to this system of linear equations. Where do the lines intersect?

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