Problem 34 - Elementary Algebra

Question

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing. $$ \left\\{\begin{array}{l} 2 x-2 y=8 \\ y=-3 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system is (7, -3).

Step 1 - Write the Equations

The given system of equations is: 1. \(2x - 2y = 8\) 2. \(y = -3\)

Step 2 - Find the Y-Intercepts and X-Intercepts

For the first equation, solve for y to make it easier to graph. \(2x - 2y = 8\) Divide every term by 2: \(x - y = 4\)Solve for y: \(y = x - 4\)For the second equation: \(y = -3\) is a horizontal line at \(y = -3\)

Step 3 - Plot the Lines on a Graph

Plot the line \(y = x - 4\) by finding the intercepts:When x = 0, \(y = -4\)When y = 0, \(x = 4\)So, the line passes through (0, -4) and (4, 0).Plot the line \(y = -3\), which is a horizontal line passing through \(y = -3\) on the y-axis.

Step 4 - Determine the Intersection Point

The point of intersection is where the two lines meet on the graph. This point represents the solution to the system of equations. Both lines meet at (7, -3).

Key Concepts

Linear Equations

Linear equations form the basis of algebra and are essential for understanding more complex mathematical concepts. A linear equation is an equation that makes a straight line when graphed. The general form is expressed as: \[ Ax + By = C \] where \( A \), \( B \), and \( C \) are constants. Each point on this line is a solution to the equation. Linear equations can have variables with the highest exponent of 1, and they don't curve.

Graphing Methods

Graphing methods involve plotting linear equations on a graph to find their solutions visually. Here is a step-by-step approach:

• Find Intercepts: Calculate where the line crosses the x-axis and y-axis.
• Plot Points: Use the intercepts to mark the line on the graph.
• Draw the Line: Connect the points to visualize the line.

By graphing both equations, you can observe where the lines intersect, giving a visual solution to the system. This method is especially useful when dealing with systems of linear equations.

Intersection Point

In graphing, the intersection point is where two lines cross each other. This point satisfies both equations simultaneously. To find the intersection:
1. Plot both equations on the same graph. 2. Identify the point where the lines meet.

In our given system, the lines intersect at \(7, -3\). This means \(x = 7 \) and \(y = -3\) satisfy both equations. The intersection point is the solution to the system of equations.

Y-Intercept

The y-intercept is the point where a line crosses the y-axis. It occurs when \(x = 0\). For example, in the equation \( y = x - 4 \), the y-intercept is \( -4 \) (when \( x = 0 \), \( y = -4 \)). Knowing the y-intercept helps in accurately plot the line on a graph. To determine the y-intercept:

• Set \( x = 0 \).
• Solve for \( y \).

For a horizontal line such as \( y = -3 \), the y-intercept is consistently \( -3 \).

X-Intercept

The x-intercept is the point where a line crosses the x-axis. It occurs when \( y = 0\). For the given equation \(y = x - 4\), we find the x-intercept by setting \( y = 0 \):

• Solve for \( x \) when \( y = 0 \).
• For instance: \(0 = x - 4 \Rightarrow x = 4\)

Knowing the x-intercept (such as \(4\) in this case) allows you to plot the line correctly. The x-intercept offers another reference point for drawing the graph accurately.