Section 10.1 - Graphs of Systems of Linear Equations

Overview

Monica Brannan works through Section 10.1 homework on graphing and interpreting systems of linear equations, demonstrating point substitution, line classification, slope comparisons, intercepts, and graph-based solutions. Examples show how intersections identify unique solutions, while coincident lines produce infinitely many solutions; a final Desmos calculation approximates a decimal-coefficient system’s solution as (0.783, -12.133).

Key takeaways

A point solves a system only if its coordinates make every equation true; one failed equation is enough to rule it out.
Equations of the form x = c graph as vertical lines, while equations of the form y = c graph as horizontal lines.
When two lines intersect once, the intersection coordinates are the system’s unique solution; coincident lines instead yield infinitely many solutions.
Equal slopes indicate parallel lines, while slopes that are negative reciprocals indicate perpendicular lines.
To find intercepts from an equation, set y = 0 for the x-intercept and x = 0 for the y-intercept.
Desmos can approximate intersections of decimal-coefficient systems, with coordinates rounded to a requested precision such as three decimal places.

Chapters

0:00 Test Candidate Points in a Two-Equation System

For the system x − y = 5 and 4x + 3y = −3, substitute each proposed point into both equations.
(3, −2) satisfies x − y = 5 but fails the second equation; (5, 0) also fails the second equation.
(2, 3) fails the first equation, so it can be rejected without checking the second.

4:46 Recognize Horizontal, Vertical, and Slanted Lines by Equation Form

A vertical line has the form x = a number, while a horizontal line has the form y = a number.
An equation containing both x and y generally describes a slanted line.
The equation 10 = −4y rearranges to y = −5/2, so its graph is horizontal.

8:56 Graph y = −x + 4 and y = x to Find Their Intersection

Rewrite x = 4 − y as y = −x + 4, and simplify 3y = 3x to y = x.
Graph the first line from y-intercept 4 with slope −1, and the second from the origin with slope 1.
The lines intersect at (2, 2), giving the system one unique solution rather than an inconsistent or dependent system.

14:39 Graph a Slope-Intercept Function and a System with y = 3

For y = (4/7)x + 1, plot the y-intercept (0, 1) and use rise 4, run 7 to locate (7, 5).
In the next system, k(x) = 3 is the horizontal line y = 3; f(x) = (9/7)x + 3 has the same y-intercept.
The graphs meet at (0, 3), so the system has exactly one solution.

21:29 Identify Perpendicular Lines and Calculate Both Intercepts

The lines with slopes 8/5 and −5/8 are perpendicular because their slopes are negative reciprocals.
For 3x − 6y = 18, set y = 0 to find the x-intercept (6, 0).
Set x = 0 to find the y-intercept (0, −3).

25:50 Verify Solutions and Graph a Vertical-Horizontal Line Pair

For x + 2y = 4 and y = −(1/2)x + 2, the points (−2, 3), (4, 0), and (3, 1/2) satisfy both equations.
Because the two equations describe the same line, the system is dependent and has infinitely many solutions.
The system y + 8 = 5 and −3 = 2x simplifies to y = −3 and x = −3/2; its horizontal and vertical lines intersect at (−3/2, −3).

34:46 Approximate a Decimal-Coefficient System with Desmos

Enter both decimal-coefficient equations into Desmos or a graphing calculator to locate their intersection.
Use the graph’s intersection readout and round each coordinate to three decimal places.
The approximate solution is (0.783, −12.133).

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Monica Brannan.