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Ch. 5 - Systems and Matrices
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 5 - Systems and MatricesProblem 41
Chapter 6, Problem 41

Determine the system of equations illustrated in each graph. Write equations in standard form.
Graph showing two intersecting lines with labeled axes and the intersection point at (12/11, 15/11).

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1
Identify the two lines and their key points from the graph. The first line passes through points (0, 7) and (-4, 0). The second line passes through points (0, 2) and (6, 0).
Find the slope of the first line using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Using points (0, 7) and (-4, 0), calculate \(m_1 = \frac{0 - 7}{-4 - 0} = \frac{-7}{-4} = \frac{7}{4}\).
Use the point-slope form of a line equation \(y - y_1 = m(x - x_1)\) with point (0, 7) and slope \(\frac{7}{4}\) to write the equation of the first line: \(y - 7 = \frac{7}{4}(x - 0)\).
Convert the first line's equation to standard form \(Ax + By = C\) by multiplying both sides to clear fractions and rearranging terms.
Repeat the process for the second line: find slope \(m_2 = \frac{0 - 2}{6 - 0} = \frac{-2}{6} = -\frac{1}{3}\), use point-slope form with point (0, 2), then convert to standard form.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Finding the Equation of a Line from Two Points

To find the equation of a line, use two points on the line to calculate the slope (m) as the change in y over the change in x. Then apply the point-slope form y - y1 = m(x - x1) to write the equation. This method is essential for determining the line's equation from graph points.
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Finding Equations of Lines Given Two Points

Converting to Standard Form of a Linear Equation

Standard form of a linear equation is Ax + By = C, where A, B, and C are integers, and A ≥ 0. After finding the slope-intercept form, rearrange terms to get all variables on one side and the constant on the other. This form is often required for system of equations problems.
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Converting Standard Form to Vertex Form

Interpreting Graphs to Identify Key Points

Graphs provide visual data such as intercepts and points where lines cross. Identifying these points accurately is crucial for writing equations. For example, x- and y-intercepts give direct points to use in slope calculations and equation formulation.
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Graphing Equations of Two Variables by Plotting Points
Related Practice
Textbook Question

Graph the solution set of each system of inequalities.

3x + 5y ≤ 15

x2 + y2 < 9

Textbook Question

Graph the solution set of each system of inequalities.

2x + y > 2

x - 3y < 6

Textbook Question

Determine the system of equations illustrated in each graph. Write equations in standard form.

3
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Textbook Question

Find each sum or difference, if possible. See Examples 2 and 3.

3
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Textbook Question

Let A=[−2403]A = \(\left\)[ \(\begin{matrix}\) -2 & 4 \\ 0 & 3 \(\end{matrix}\) \(\right\)] and B=[−6240]B = \(\left\)[ \(\begin{matrix}\) -6 & 2 \\ 4 & 0 \(\end{matrix}\) \(\right\)]. Find each of the following.

2A

Textbook Question

Find the partial fraction decomposition for each rational expression. See Examples 1–4. (4x2 - 3x - 4)/(x3 + x2 - 2x)