Problem 23 - Algebra 1

Question

Graph and check to solve the linear system. $$ \begin{aligned} &2 x-3 y=9\\\ &x=-3 \end{aligned} $$

Step-by-Step Solution

Answer

The solution to the given system of linear equations is the point (-3,-5).

Step 1: Representing Equation 1 Graphically

The first equation \(2x - 3y = 9\) can be rewritten in slope-intercept form as \(y = \frac{2}{3}x - 3\). The slope of this line is \frac{2}{3} and it intersects the y-axis at y = -3.

Step 2: Representing Equation 2 Graphically

The second equation is a vertical line represented by \( x = -3\). There is no y-intercept since the line is parallel to the y-axis.

Step 3: Point of Intersection and Solution to the System

The solution to the system can be found at the point where the two lines intersect. Since the second equation is \(x = -3\) and it is a vertical line, it intersects the first line at the x-coordinate -3. Therefore, substitute x = -3 into the first equation to find the corresponding y-coordinate. \(y = \frac{2}{3}*(-3) - 3 = -5\). So, the system of equations intersect at the point (-3,-5).

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is foundational for graphing linear equations. The slope-intercept form is written as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) denotes the y-intercept, which is the point where the line crosses the y-axis.

For example, if we have a linear equation \( 2x - 3y = 9 \), to convert this to slope-intercept form, we solve for \( y \). First, we would move the \( 2x \) to the other side, giving us \( -3y = -2x + 9 \). Then, by dividing each term by the coefficient of \( y \) which is -3, we would get \( y = \frac{2}{3}x - 3 \). This tells us immediately that the slope of the line, \( m \), is \( \frac{2}{3} \) and the y-intercept, \( b \), is -3. This quick transformation allows for easy graphing and understanding of the line's behavior on a coordinate plane.

Graphing Linear Equations

To visually solve a linear system, graphing each equation on the same set of axes is crucial. After converting to slope-intercept form, we graph the lines by starting at the y-intercept and using the slope to find other points.

The slope \( m \) as a fraction \( \frac{rise}{run} \) tells us how to move from one point to another on the line. A positive slope means the line ascends from left to right, while a negative slope descends. In our example with \( y = \frac{2}{3}x - 3 \), starting at the y-intercept (0, -3), we 'rise' up 2 units and 'run' to the right 3 units to locate another point on the line, and we continue this pattern to draw the entire line.

For a vertical line like \( x = -3 \), there's no slope to consider; we simply draw a straight line parallel to the y-axis that passes through the x-coordinate of -3. Similarly, horizontal lines have a slope of zero and are graphed parallel to the x-axis.

Point of Intersection

The point of intersection of two lines is where they cross each other on the coordinate plane; it is the solution to the system of linear equations represented by the lines. For our equations, once graphed, we look for the point where both lines meet.

In cases where one line is vertical (such as \( x = -3 \)) and the other has a slope (like \( y = \frac{2}{3}x - 3 \)), their intersection is straightforward to find because the x-value of the point is given directly by the vertical line's equation, and the y-value can be found by substituting this x-value into the other equation. As performed in our earlier solution, substituting \( x = -3 \) into the second equation gave us the y-coordinate (-5), leading us to the point of intersection (-3, -5).

This point is the only pair of x and y values that satisfies both equations simultaneously, meaning it is where the physical paths of both equations' graphs cross, and symbolically, it is the solution to the system of equations.