Problem 113 - College Algebra Essentials

Question

Find the coefficients that must be placed in each shaded area so that the function's graph will be a line satisfying the specified conditions. \(x+\quad y-12=0 ; x\) -intercept \(=-2 ; y\) -intercept \(=4\)

Step-by-Step Solution

Answer

The coefficients that make the function's graph a line satisfying the given conditions are 1 for \(x\), -1/3 for \(y\), and 12 for the constant term.

Step 1: Finding the x-intercept

The \(x\)-intercept of a line is found by setting \(y = 0\) in the equation and solving for \(x\). In this case, if \(x + y - 12 = 0\) when \(y = 0\), then \( x - 12 = 0\). Therefore, \(x = 12\). However, it is given that the \(x\)-intercept must be \(-2\). Therefore, a coefficient of \(-1\), instead of \(1\), should multiply \(x\) to yield \(-2\) for \(x\) instead of \(12\). The equation is then transformed into \(-x + y - 12 = 0\)

Step 2: Finding the y-intercept

The \(y\)-intercept of a line is found by setting \(x = 0\) in the equation and solving for \(y\). By plugging \(x = 0\) into the equation, we get \(-x + y - 12 = 0\), solving for \(y\) gives \(y = 12\). However, it is given that the \(y\)-intercept must be \(4\). Therefore, a coefficient of \(\frac{1}{3}\), instead of \(1\), should multiply \(y\) to yield \(4\) for \(y\) instead of \(12\). Multiplying the whole equation by \(3\) then gives \(-3x + y - 36 = 0\)

Step 3: Finding the constant term

From the above steps, we found that the equation should be \(-3x + y - 36 = 0\) to satisfy the given conditions. However, the constant term in the original equation is \(-12\), not \(-36\). Therefore, we should divide the whole equation by \(-3\) to get the desired constant term of \(-12\). The final equation is \(x - \frac{1}{3}y + 12 = 0\)

Key Concepts

Understanding the X-Intercept

The x-intercept of a linear equation is the point where the line crosses the x-axis. At this point, the y-coordinate is always zero. To find the x-intercept, we set the equation equal to zero and solve for the x variable.

For example, in the equation \[x + y - 12 = 0\], to find the x-intercept, we set y to zero and solve for x.

• Set \[y = 0\] in the equation.
• The equation becomes \[x - 12 = 0\].
• Solving this equation gives \[x = 12\].

However, if the x-intercept is given as \[-2\], we recognize a need to adjust the coefficients. The incorrect x-intercept suggests that the original coefficient of x needs to be replaced.

Discovering the Y-Intercept

The y-intercept is the point where the line intersects the y-axis, meaning at this point, the x-coordinate is zero. To find the y-intercept in a linear equation, we substitute zero for x and solve for y.

Using the modified equation, \[-x + y - 12 = 0\]:

• Set \[x = 0\] in the equation.
• Replace this in the equation to get \[y - 12 = 0\].
• Solving it provides \[y = 12\].

If the given y-intercept is \[4\], a recalibration of the y coefficient is necessary. Transforming variables' coefficients will guide us to the correct intercept, using multiplication or division to adjust the equation.

Identifying Coefficients

Coefficients in a linear equation are the numbers that multiply the variables, influencing the slope and intercepts of the line. They are crucial to shaping the behavior and position of the line on the graph.

In the equation, \[-3x + y - 36 = 0\], the coefficients are:

• \[-3\] for \[x\], affecting the slope.
• \[1\] for \[y\], influencing vertical displacement.
• The constant term here is \[-36\], impacting line positioning vertically.

Should the intercepts be incorrect, these coefficients need tweaking to align them properly with specified intercept conditions. Reexamining and modifying coefficients are essential steps towards resolving discrepancies in intercept analysis.

Solving Linear Equations

Solving a linear equation involves finding the values of variables that make the equation true. It's a process of isolating variables, often requiring adjustments in coefficients or constants to achieve desired intercepts and solutions.

In solving \[x - \frac{1}{3}y + 12 = 0\], we:

• Focus on isolating each variable by algebraic manipulation such as adding, subtracting, or multiplying both sides by numbers.
• Reexamine y-intercept and x-intercept with our set initial conditions.

Understanding and applying these techniques ensures our linear equation reflects the requirements of specific intercepts. It's a key skill in algebra, fostering deeper comprehension of how linear equations graphically represent relations between variables.