Problem 91 - Introductory and Intermediate Algebra for College Students 4th

Question

Find the coefficients that must be placed in each shaded area so that the equation's graph will be a line with the specified intercepts. \(\square x+\square y=10 ; x\) -intercept \(=5 ; y\) -intercept \(=2\)

Step-by-Step Solution

Answer

The coefficients that must be placed in each shaded area so that the equation's graph will be a line with x-intercept = 5 and y-intercept = 2 are 2 (for x) and 5 (for y). So, the equation is \(2x + 5y = 10\).

Step 1: Form equations using the intercepts

First, use the x-intercept and y-intercept to form two separate equations. For the x-intercept, y-value is 0, and for the y-intercept, x-value is 0.\nSo, for x-intercept = 5, the equation is \( \square \cdot 5 + \square \cdot 0 = 10 \).\nFor y-intercept = 2, the equation is \( \square \cdot 0 + \square \cdot 2 = 10 \). This gives us two equations: \(5 \square = 10\) and \(2 \square = 10\).

Step 2: Solve the equations

Now, solve these equations to find the coefficients. For \(5 \square = 10\), divide both sides by 5 to get: \(\square = 2\). For \(2 \square = 10\), divide both sides by 2 to get: \(\square = 5\).

Step 3: Identify the coefficients

The solutions for the two equations give the coefficients. So, the coefficient for x is the solution from the equation formed with the x-intercept, which is 2. The coefficient for y is the solution from the equation formed with the y-intercept, which is 5.

Key Concepts

X-Intercept

The x-intercept of a line is the point where the line crosses the x-axis. This means that at the x-intercept, the y-coordinate is always zero. Finding the x-intercept helps us understand one of the specific points the line passes through on the graph.

To determine the x-intercept for the equation of a line, you substitute

• y = 0 into the equation, and
• solve for x.

For example, with the equation \( ax + by = 10 \), if the x-intercept is 5, you substitute the y value as 0, simplifying the equation to \( ax + 0 = 10 \), or \( ax = 10 \). Solving this gives the value of x, confirming the x-intercept.

Understanding the x-intercept is crucial because it tells you exactly where the line will pass through the x-axis on a graph, a fundamental point in plotting any linear equation.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis. At the y-intercept, the x-coordinate is always zero. This is another special point used to define a line graphically.

To find the y-intercept:

• Set x = 0 in the equation, and
• solve for y.

For the equation \( ax + by = 10 \), given that the y-intercept is 2, when x is set to 0, the equation becomes \( 0x + by = 10 \) or \( by = 10 \). Solving this equation will give the value of y, confirming the y-intercept.

The y-intercept provides valuable information, helping you visualize where the line starts on the y-axis. It is the second crucial point needed to sketch a line within a coordinate plane.

Coefficients

Coefficients are the numerical factors that multiply the variables in an equation. In a linear equation like \( ax + by = c \), \( a \) and \( b \) are the coefficients of x and y, respectively.

Determining coefficients is vital for defining the slope and position of a line in a plane. In the context of x and y intercepts:

• The coefficient of x, derived with the x-intercept, indicates the "steepness" relative to the x-axis.
• The coefficient of y, derived with the y-intercept, reveals the "steepness" relative to the y-axis.

In our original exercise, the coefficients were determined by solving the two simpler equations resulting from substituting intercepts back into the original equation. Thus, finding \( a = 2 \) and \( b = 5 \) provides key details to reconstruct the original linear equation's graph.

Graph of a Line

The graph of a line represents the set of all solutions or points that satisfy its linear equation. It is straight, continuous, and extends infinitely in both directions, unless specified otherwise.

When plotting a line, you often need:

• The x-intercept
• The y-intercept
• Coefficients
• The slope, which can be derived from the coefficients

These components help draw the exact path the line will take across a graph.

Using the intercepts,

• The x-intercept point is \((x, 0)\)
• The y-intercept point is \((0, y)\)

Plotting these two points on a graph and connecting them with a straight line provides a visual representation of the linear equation. The coefficients inform the angle and direction of this line, allowing you to understand the relationship between the variables visually. This graphical representation is a valuable tool for interpreting data and establishing patterns.