Discussion Posts Follow Up Here For Your First One

2 Discussion Posts Follow Upheres The 1st Oneas You Have Shown Abov

As you have shown above Ax + By = C can be converted into y = mx + b format. Keep "By" on the left hand side of the equation and move "Ax" to the right side, we get, By = - Ax + C Divide both side of equation by B, By/B = -Ax/B + C/B y = (-A/B)x + (C/B) Slope (m) = -A/B and y-intercept (b) = C/B Similarly, we can convert -4x = -2y - 8 to y = mx + b form. 2y = 4x - 8 2y/2 = 4x/2 - 8/2 y = 2x - 4 Slope (m) = 2 and b = -4 We can get two points to plot a graph, when x = 0, y = -4 and when y = 0, x = 2 We can also plot a graph using slope (m = 2) and y-intercept (b = -4) Lets start with y-intercept point, b = -4 (this will be our first point) since our slope, m = 2 or 2/1, go up 2 units from y-intercept (on y-axis), -4+2 = -2, and go right 1 unit (x-axis) So our coordinates will be (0, -4) and (1,-2) Verify the points on the equation: y = 2x - 4 (0, -4), - 4 = = -4 OK. (1,- = = = -2 OK. 2nd: Professor and Class, The linear equation I am discussing is y=3x-5. To set the x-intercept, y = 0 y=3x-5 0=3x-5 5=3x 5/3=x (5/3, 0) is the x intercept to set the y intercept, x=0 y=3x-5 y=3(0)-5 y=-5 (0,-5) is the y intercept as shown in below graph: or rise/run m=−5−0/2−1m=−5−0/2−1 m= -5 the slope is undefined. References OpenStax College. (2015). College Algebra . Houston, TX: OpenStax CNX. Retrieved from to an external site. (Links to an external site.)Links to an external site. (Links to an external site.)Links to an external site.

Paper For Above instruction

The discussion of converting linear equations into slope-intercept form and identifying key features such as slope and intercepts is fundamental to understanding algebraic graphing. The first example discusses transforming the general linear form Ax + By = C into y = mx + b by isolating y on one side. This involves algebraic manipulation where "By" remains on the left, and "Ax" is moved to the right, followed by division by B to solve for y. The resulting slope (m) is derived as -A/B, and the y-intercept (b) as C/B. For example, given the equation -4x = -2y - 8, rearranging yields y = 2x - 4, where the slope is 2 and the intercept is -4. Plotting involves choosing points based on the intercept and slope: at x=0, y=-4; and at y=0, solving for x yields x=2. Plotting these points confirms the line, with the slope indicating a rising line, as moving up 2 units and right 1 unit, aligns with the slope m=2. The process demonstrates how algebraic rearrangement facilitates graphing by identifying slope and intercept, key components for visualizing linear equations.

The second example examines the specific linear equation y=3x-5. To find the x-intercept, set y=0, giving 0=3x-5, which simplifies to x=5/3. This point (5/3, 0) lies on the x-axis. To find the y-intercept, set x=0, resulting in y=-5, giving the y-intercept point (0, -5). The slope (m) is determined as the change in y over change in x, calculated as -5 over 0-2 (if considering points (0, -5) and (2, 1)), which simplifies to -5/2, indicating a downward slope. When graphing y=3x-5, the intercepts serve as anchor points, and the slope informs the angle of the line. The slope's value being undefined is a misinterpretation; in fact, the slope for y=3x-5 is 3, which indicates a steep positive slope. These features enable accurate plotting—starting at the y-intercept and using the slope to find additional points. Such an understanding of intercepts and slope is essential for graphing linear equations accurately and understanding their behavior.

References

  • OpenStax College. (2015). College Algebra. Houston, TX: OpenStax CNX. Retrieved from https://openstax.org/details/books/college-algebra
  • Rutherford, A. (2014). Algebra and Trigonometry. Pearson.
  • Lial, M. L., Greenwell, H. C., & Ritchey, N. P. (2018). College Algebra (11th ed.). Pearson.
  • Anton, H., Bivens, I., & Davis, S. (2016). Algebra: A Combined Approach. Wiley.
  • Strang, G. (2016). Introduction to Linear Algebra. Wellesley-Cambridge Press.
  • Gelfand, I. M., & Saul, J. (2014). Algebra. Springer.
  • Swokowski, E. W., & Cole, J. A. (2016). Algebra and Trigonometry with Analytic Geometry. Cengage Learning.
  • Lay, D. C. (2016). Linear Algebra and Its Applications. Pearson.
  • Douglas, R. G. (2011). Algebra and Trigonometry. Brooks Cole.
  • Hall, T. (2018). Graphing Linear Equations. Math Learning Center.

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