Problem 29 - Elementary Algebra

Question

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing. $$ \left\\{\begin{array}{l} -2 x+3 y=3 \\ x+3 y=12 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is the point where the lines intersect.

Step 1 - Rewrite Equations in Slope-Intercept Form

First, solve each equation for y in terms of x to get them into slope-intercept form, y = mx + b.

Step 1A - Solve First Equation

For the equation \(-2x + 3y = 3\), add \(2x\) to both sides: \[3y = 2x + 3\] Next, divide by 3: \[y = \frac{2}{3}x + 1\]

Step 1B - Solve Second Equation

For the equation \(x + 3y = 12\), subtract \(x\) from both sides: \[3y = -x + 12\] Next, divide by 3: \[y = -\frac{1}{3}x + 4\]

Step 2 - Plot the Equations on a Graph

Plot the lines \(y = \frac{2}{3}x + 1\) and \(y = -\frac{1}{3}x + 4\) on the graph. Start by identifying the y-intercept (b) and then use the slope (m) to plot additional points.

Step 2A - Plot the First Line

For \(y = \frac{2}{3}x + 1\), the y-intercept is 1. From this point, use the slope \(\frac{2}{3}\) to plot more points: go up 2 units and right 3 units to find another point.

Step 2B - Plot the Second Line

For \(y = -\frac{1}{3}x + 4\), the y-intercept is 4. From this point, use the slope \(-\frac{1}{3}\) to plot more points: go down 1 unit and right 3 units to find another point.

Step 3 - Determine the Intersection Point

Observe where the two lines intersect. The coordinates of the intersection point are the solution to the system of equations.

Step 4 - Verify the Solution

Substitute the intersection point's coordinates into the original equations to verify they work in both.

Step 4A - Verify with First Equation

Substitute the intersection point into \(-2x + 3y = 3\).

Step 4B - Verify with Second Equation

Substitute the intersection point into \(x + 3y = 12\)

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is essential for graphing. It is written as:
\[ y = mx + b \]
\(*m*\) represents the slope of the line, and \(*b*\) is the y-intercept. The slope indicates how steep the line is and in which direction it slants. For instance, a slope of \[ \frac{2}{3} \] means that for every 3 units you move horizontally to the right, you move 2 units vertically up. In the exercise, the equations are converted into this form before graphing them:

• First equation: \[ y = \frac{2}{3}x + 1 \]
• Second equation: \[ y = -\frac{1}{3}x + 4 \]
  Converting to slope-intercept form makes it easier to identify the y-intercept and graph the line accurately. Start plotting from the y-intercept (the point on the line where x = 0) and then use the slope to find the next points.

Linear Equations

Linear equations graph as straight lines on a coordinate plane. Each equation can be expressed in various forms, with slope-intercept being one of the most useful for graphing. Consider the given system of equations:

• \[ -2x + 3y = 3 \]
• \[ x + 3y = 12 \]
  Converting these to slope-intercept form makes it simpler to understand how they will graph:
  • First equation: \[ y = \frac{2}{3}x + 1 \]
  • Second equation: \[ y = -\frac{1}{3}x + 4 \]
    Notice each equation represents a line, and every point on a line is a solution to the equation. To solve the system graphically, both lines are plotted on the same graph, and their intersection represents the simultaneous solution.

Intersection Point

When graphing systems of linear equations, the point where two lines intersect reveals the solution to the system. To find this intersection:

• First, plot each equation on the graph starting with their y-intercepts
• Next, follow each slope to mark additional points
  For the system
  • \[ y = \frac{2}{3}x + 1 \]
  • \[ y = -\frac{1}{3}x + 4 \]
  
  Plotting these reveals an intersection at a specific point, say (3, 3). This intersection point is the solution, implying x=3, y=3 satisfies both equations, meaning both lines meet at this point on the graph. This visual method offers an intuitive way to solve the system.

Verify Solution

Verification ensures the solution obtained from graphing is accurate. To verify, substitute the intersection point's coordinates into the original equations to see if they hold true. For example, with the solution (3, 3):

• For \[ -2x + 3y = 3 \] \[ -2(3) + 3(3) = -6 + 9 = 3 \]
• For \[ x + 3y = 12 \] \[ 3 + 3(3) = 3 + 9 = 12 \]
  Both equations are true with this point, confirming (3, 3) as the correct solution. Verifying each step reassures us that the intersection point truly satisfies both original equations. This step is crucial to avoid errors in graphing and solution identification.