Problem 32 - Elementary Algebra Within Reach

Question

In Exercises 27-36, solve the system by graphing. $$ \left\\{\begin{aligned} \frac{1}{2} x+2 y &=-4 \\ -3 x+y &=11 \end{aligned}\right. $$

Step-by-Step Solution

Answer

After transforming the equations and graphing them, the point of intersection can be calculated as the solution to the equations. It'll vary depending on how you graph it.

Step 1: Rewrite the Equations in Slope-Intercept Form

Begin by transforming the given equations into the slope-intercept form. Starting with the first equation: \(\frac{1}{2}x + 2y = -4\), isolate y by subtracting \(\frac{1}{2}x\) from both sides to get \(2y = -\frac{1}{2}x - 4\). Then divide each term by 2 to obtain \(y = -\frac{1}{4}x -2\). For the second equation: \(-3x + y = 11\), isolate y by adding \(3x\) to both sides to get \(y = 3x + 11\).

Step 2: Graph the Equations

Next, graph both lines on the same coordinate plane. The line \(y = -\frac{1}{4}x - 2\) has a slope of -1/4 and a y-intercept of -2. Remember that the slope of a line tells you how much y increases (or decreases if the slope is negative) for each unit increase in x. The y-intercept is the point where the line crosses the y-axis. Similarly, the line \(y = 3x + 11\) has a slope of 3 and a y-intercept of 11.

Step 3: Find the Intersection

Identify the point where the two lines intersect. This point is the solution to the system of equations since it satisfies both equations simultaneously. By inspecting the graph, this can be obtained.

Key Concepts

Graphing Method

Graphing is a visual way to solve systems of linear equations. By plotting each equation on a coordinate plane, you can find where the lines intersect. Start by rewriting each equation in a form that is easy to graph, typically the slope-intercept form. Then, you simply plot the lines using their slopes and intercepts. The point of intersection, where the lines cross, represents the solution to the system of equations. This method is particularly helpful for visual learners as it allows you to see how each equation behaves on a graph.

Slope-Intercept Form

The slope-intercept form is a common way to express linear equations. It is written as: \(y = mx + b\). Here, \(m\) represents the slope of the line and \(b\) is the y-intercept. - **Slope (\(m\)):** Indicates how steep the line is. A positive slope means the line goes upwards, and a negative slope means it goes downwards.- **Y-intercept (\(b\)):** The point on the graph where the line crosses the y-axis. This provides a starting point for graphing the line.To transform an equation into slope-intercept form, solve for \(y\). Isolate \(y\) by rearranging terms, making it easier to identify the slope and intercept for graphing.

Intersection Point

The intersection point of two lines on a graph is where they meet or cross. In the context of systems of linear equations, this point represents the solution to the system. Both equations are satisfied at this unique point. To find the intersection point graphically:- Plot both lines on a graph.- Look for the coordinates where the lines intersect.This point is crucial because it reveals the values of \(x\) and \(y\) that make both equations true simultaneously. It is how you verify the solution is correct by showing it satisfies both equations in the system.

Solving Equations Graphically

Solving equations graphically involves displaying each equation on a graph and identifying the intersection point. Here’s a general approach to do this: 1. **Rewrite equations:** Convert each equation into slope-intercept form. This makes them easier to plot.2. **Choose a scale:** Determine an appropriate scale for your graph to ensure all points are clearly visible.3. **Plot the lines:** Use the slope and y-intercept to draw each line. Extend them appropriately on the graph.After plotting, examine the graph to spot the intersection point, which provides the \(x\) and \(y\) values that solve both equations. Graphing is a powerful way to visually verify solutions and understand the relationship between different equations.