Problem 91 - Introductory Algebra for College Students

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 for x and y are 2 and 5 respectively, resulting in the equation \(2x + 5y = 10\).

Step 1: Determine the x-intercept

In order to find the x-intercept, we substitute \(y = 0\) into the equation. We obtain \(\square x + \square * 0 = 10\) which simplifies to \(\square x = 10\). Since the x-intercept is given to be 5, we can replace \(x\) with 5 to get \(\square * 5 = 10\). From this we find that the coefficient of \(x\) (in the square) is 2.

Step 2: Determine the y-intercept

For the y-intercept, we substitute \(x = 0\) into the equation. We obtain \(\square * 0 + \square y = 10\) which simplifies to \(\square y = 10\). Given the y-intercept is 2, we can replace \(y\) with 2 to get \(\square * 2 = 10\). From this we can see that the coefficient of \(y\) (in the square) is 5.

Step 3: Write the equation

Now that we have the coefficients of \(x\) and \(y\), we can write the linear equation as \(2x + 5y = 10\).

Key Concepts

Finding X-Intercept

The x-intercept of a linear equation is the point at which the graph of the equation crosses the x-axis. This occurs when the y-coordinate is zero. To find the x-intercept, you simply set the y-value to zero and solve for x.

For example, let's say we have an incomplete equation \( \square x + \square y =10 \) and we know the x-intercept is 5. By substituting y with 0, the equation becomes \( \square x = 10 \). Knowing that the x-intercept, which is the x-value when y equals zero, is 5, we can determine that \( \square * 5 = 10 \). Solving for the square gives us 2, indicating that the x-coefficient is 2.

Finding Y-Intercept

Conversely, the y-intercept is where the line crosses the y-axis, which is when x equals zero. To find this value, we set the x-value to zero in the equation and solve for y.

When we apply this process to our incomplete equation \( \square x + \square y =10 \) with a known y-intercept of 2, putting x as zero simplifies to \( \square y = 10 \). Then, \( \square * 2 = 10 \) allows us to figure out that the missing coefficient before y should be 5. Thus, the coefficient of y is determined to be 5.

Determining Coefficients

Coefficients in a linear equation are the numerical factors that multiply the variables. They are essential as they determine the slope of the line, which represents how steep the line is on a graph.

In our example, the coefficients are found by understanding the relationships given by the intercepts. Once we placed 0 in for y and then x respectively, and used the known intercepts to solve the equations, we determined the coefficients of x and y to be 2 and 5. Hence, the completely formulated linear equation is given by \(2x + 5y = 10\).

Linear Equation Graphing

Graphing a linear equation involves plotting points that satisfy the equation and then drawing a line through these points. With the equation \(2x + 5y = 10\), you can find at least two points using the intercepts—(5, 0) for the x-intercept and (0, 2) for the y-intercept.

By marking these points on a coordinate system and drawing a straight line through them, you create the graph of the equation. The slope of the line can be calculated by the coefficient ratio, which is the y-coefficient divided by the x-coefficient, in this case \(\frac{5}{2}\).