Solving Systems of Equations by Graphing

Slope-Intercept Form and Point Identification

• Standard Form: The equations in this lesson are presented in slope-intercept form, which is defined as $y = mx + b$.

• Key Components of the Form:     * $m$ represents the slope of the line.     * $b$ represents the $y$-intercept, specifically the point $(0, b)$ where the line crosses the $y$-axis.

• Methodology for Identifying Points:     * First, identify the $y$-intercept directly from the constant $b$.     * Second, use the slope ($m = \frac{\text{rise}}{\text{run}}$) to find a second point. By starting at the $y$-intercept and moving to the right by one unit (a run of $1$), the $y$-coordinate changes by the value of the slope.

Detailed Point Calculation for Equation 1: $y = -2x + 7$

• Identification of Constants:     * The slope ($m$) is $-2$.     * The $y$-intercept ($b$) is $7$.

• Initial Point: The line passes through $(0, 7)$.

• Calculation of the Second Point:     * Using the slope of $-2$, we increment the $x$-coordinate by $1$ and the $y$-coordinate by $-2$.     * Formula: $(0 + 1, 7 - 2)$.     * Resulting Point: $(1, 5)$.

• Graphing Requirement: These two points, $(0, 7)$ and $(1, 5)$, are sufficient to draw the first line accurately.

Detailed Point Calculation for Equation 2: $y = 5x - 7$

• Identification of Constants:     * The slope ($m$) is $5$.     * The $y$-intercept ($b$) is $-7$.

• Initial Point: The line passes through $(0, -7)$.

• Calculation of the Second Point:     * Using the slope of $5$, we increment the $x$-coordinate by $1$ and the $y$-coordinate by $5$.     * Formula: $(0 + 1, -7 + 5)$.     * Resulting Point: $(1, -2)$.

• Graphing Requirement: These two points, $(0, -7)$ and $(1, -2)$, allow for the accurate graphing of the second line.

Solving the System through Graphing

• The Solution Concept: The solution to a system of linear equations is the point where the two lines intersect on the coordinate plane. This point represents the values of $x$ and $y$ that satisfy both equations simultaneously.

• Observed Intersection: Upon graphing the lines defined by $y = -2x + 7$ and $y = 5x - 7$, the lines meet at a specific coordinate.

• Intersection Coordinates: $(2, 3)$.

• Final System Solution:     * $x = 2$     * $y = 3$

Related Concepts and Content Extensions

• Alternative Fractions in Systems: Similar methods are applied to equations featuring fractional slopes, such as:     * $y = \frac{7}{5}x - 5$     * $y = \frac{3}{5}x - 1$

• Solution Types: Lessons distinguish between results that provide exact solutions (integer coordinates) and approximate solutions, where the intersection may not fall precisely on the grid intersections.

• Skills Inventory: This material is categorized under "Graphing the equations" and "Solving the system" within the Level 6 algebra skills of the Khan Academy curriculum.

To find points from an equation given in slope-intercept form $y = mx + b$, follow these steps:

1. Identify the Constants:

   • Determine the slope ($m$) and the $y$-intercept ($b$) from the equation.

     • For example, in the equation $y = -2x + 7$:

     • Slope ($m$) = $-2$

     • $y$-intercept ($b$) = $7$.

2. Initial Point:

   • Plot the initial point on the graph at $(0, b)$.

     • Here, the point is (0, 7)$.

3. Using the Slope to Find the Second Point:

   • Use the slope to move from the initial point. The slope is defined as ext{rise/run}$.</p></li><li><p>For$m = -2$, this means move down 2 units (rise) and 1 unit to the right (run).</p><ul><li><p>From the point$(0, 7)$:</p></li><li><p>New point:</p><ul><li><p>$x = 0 + 1 = 1$</p></li><li><p>$y = 7 - 2 = 5$</p></li></ul></li><li><p>This gives the point$(1, 5)$.</p></li></ul></li></ul></li><li><p><strong>Graphing the Points</strong>:</p><ul><li><p>The two points to graph are$(0, 7)$and$(1, 5)$. Draw a straight line through these points to represent the equation.</p></li></ul></li><li><p><strong>Repeat for Other Equations</strong>:</p><ul><li><p>For another equation, e.g.,$y = 5x - 7$:</p><ul><li><p>Identify slope and intercept:</p></li><li><p>Slope ($m$) =$5$,$y$-intercept ($b$) =$-7$.</p></li><li><p>Initial point:$(0, -7)$.</p></li><li><p>Second point: Move up 5 units and right 1 unit. This gives us$(1, -2)$.</p></li><li><p>Graph the points$(0, -7)$and$(1, -2)$</p></li></ul></li></ul></li><li><p><strong>Solving Systems through Graphing</strong>:</p><ul><li><p>After graphing both lines, find their intersection point, which represents the solution to the system of equations.</p></li></ul></li></ol><p>In this case, graphing$y = -2x + 7$and$y = 5x - 7$results in them intersecting at$(2, 3)$, providing the solution where$x = 2$and$y = 3$$.