Problem 39 - Introductory and Intermediate Algebra for College Students 4th

Question

Use intercepts and a checkpoint to graph each equation. $$2 x-3 y=-11$$

Step-by-Step Solution

Answer

The graph of the linear equation \(2x - 3y = -11\) passes through the points (5.5, 0) and (0, 3.67), the x-intercept and y-intercept respectively. Our selected checkpoint (1,1) did not fit into our equation and does not line up.

Step 1: Find the x-intercept

To find the x-intercept, set \(y = 0\) in the equation and solve for \(x\). Thus, we get \(2x = 11\), hence \(x = \frac {11}{2} = 5.5\). So, the x-intercept is at point (5.5, 0).

Step 2: Find the y-intercept

To find the y-intercept, set \(x = 0\) in the equation and solve for \(y\). Thus, we get \(-3y = -11\), hence \(y = \frac {11}{3} = 3.67\). So, the y-intercept is at point (0, 3.67).

Step 3: Select a checkpoint

Let's choose the point (1, 1) as a check point and substitute these values into our equation. If both sides of the equation match after substituting, then our selected point is correct. Substituting we get \(2(1) - 3(1) = 2 - 3 = -1 \neq -11\), and hence our selected point (1,1) does not line up.

Step 4: Construct the graph

Based on the obtained intercepts, we construct a line passing through the points (5.5, 0) and (0, 3.67). Keep in mind that the checkpoint (1,1) did not fit into our equation.

Key Concepts

Intercepts

An intercept is a crucial point in graphing linear equations, as it is where the line intersects either the x-axis or the y-axis. When we talk about the x-intercept:

• We are interested in the point where the line crosses the x-axis.
• At this point, the value of \(y\) is always zero.
• So, to find it, we set \(y = 0\) and solve for \(x\).

For the y-intercept:

• It is the point where the line crosses the y-axis.
• At this location, the value of \(x\) is zero.
• Thus, we set \(x = 0\) and solve for \(y\).

In the given equation \(2x - 3y = -11\):* The x-intercept calculation led us to \( x = 5.5 \) so the intercept is (5.5, 0).* For y-intercept, solving gives \( y = 3.67 \), hence our intercept point is (0, 3.67). These intercepts are essential for drawing the line confidently on a graph, providing clear points to plot initially.

Coordinate Geometry

Coordinate geometry, often called analytic geometry, is a branch of mathematics that translates geometric figures into an algebraic format using coordinates. The x-y plane:

• Is a coordinate grid where each point is defined by a pair of numbers (x, y).
• The horizontal axis is the x-axis, and the vertical is the y-axis.

When solving linear equations, coordinate geometry helps us understand:

• How to plot points like (5.5, 0) and (0, 3.67) using the intercepts found.
• The slope of the equation: it determines the steepness and direction of the line.
• Visualizing and solving geometric problems algebraically.

This way, every algebraic solution can have a geometric representation, making it easier to interpret and solve many real-life problems. Here, the calculated intercepts can be marked on the graph to get a visual representation of the equation \(2x - 3y = -11\).

Linear Equations Solution

Solving linear equations, like \(2x - 3y = -11\), involves finding values of variables that satisfy the equation, typically expressed in two dimensions as a straight line. Here's how it's done:

• Identify and compute intercepts: which are the foundational points to begin graphing.
• Use a checkpoint: a point not on the intercepts to confirm the accuracy of your line.
• Graph the line: connect the intercepts to illustrate the solution on a coordinate plane.

After plotting intercepts (5.5, 0) and (0, 3.67), draw a line through these points. This line represents all possible (x, y) pairs that satisfy the equation. If needed, adjust slope calculation or use another method like slope-intercept form \(y = mx + c\) for cross-verification. Using these methods ensures the solution is both accurate and visually interpretable. As demonstrated in this equation, problem-solving with linear equations is straightforward with practice and ensures an accurate graphical depiction of solutions.