Problem 93 - Algebra for College Students

Question

Find the equation of line l in each case and then write it in standard form with integral coefficients. Line \(l\) goes through \((2,5)\) and is parallel to the \(x\) -axis.

Step-by-Step Solution

Answer

The equation of the line is 0x + 1y = 5.

Step 1 - Identify the properties of the line

Since the line is parallel to the x-axis, the y-coordinates of all points on the line are the same. This implies that the slope of the line is 0.

Step 2 - Determine the equation in slope-intercept form

The general form of a line parallel to the x-axis can be directly written as y = k, where k is the constant that gives the y-coordinate of the points on the line. In this case, since the line passes through the point (2, 5), the equation is y = 5.

Step 3 - Convert to standard form

The standard form of a linear equation is Ax + By = C. For the equation y = 5, rewrite it as 0x + 1y = 5.

Step 4 - Verify integral coefficients

Ensure the coefficients A, B, and C are integers. In this case, they are A = 0, B = 1, and C = 5, which are already integers.

Key Concepts

slope-intercept form

The slope-intercept form of a line is one of the most common ways to write the equation of a line. It is given by the formula \(y = mx + b\), where:

• \(m\) is the slope of the line.
• \(b\) is the y-intercept, the point where the line crosses the y-axis.

The slope \(m\) tells you how steep the line is. It represents the change in y for a one-unit change in x. For example, if the slope is 2, the y-value increases by 2 every time the x-value increases by 1.
In the step-by-step solution, the line was parallel to the x-axis, meaning its slope is 0. So the equation simplifies to \(y = k\), where \(k\) is a constant. In our case, since the line passes through the point \((2, 5)\), the equation becomes \(y = 5\). This is the slope-intercept form for our specific line.

standard form

The standard form of a line's equation is written as \(Ax + By = C\), where:

• \(A\), \(B\), and \(C\) are integers.
• \(A\) and \(B\) are not both zero.

The standard form is useful because it provides a straightforward way to identify the coefficients and to graph linear equations.
To convert from slope-intercept form to standard form, we use algebraic manipulation. For our line, which has the equation \(y = 5\), we can rewrite this in standard form. Since there is no x-term, we consider it as \(0x + 1y = 5\). This gives us the standard form \(0x + 1y = 5\), where \(A = 0\), \(B = 1\), and \(C = 5\). All coefficients \(A\), \(B\), and \(C\) are integers, making this a valid standard form of the line's equation.

parallel lines

Parallel lines are lines in the same plane that never intersect. They always maintain the same distance from each other. For two lines to be parallel, they must have the same slope.
In coordinate geometry, if two lines have the slopes \(m_1\) and \(m_2\), they are parallel if \(m_1 = m_2\). This means that the lines rise and run in the exact same way, never converging or diverging.
In our exercise, the line given in the problem is parallel to the x-axis. This means it has a slope of 0, just like the x-axis itself. Since the line is horizontal and does not tilt up or down, it remains at a constant y-value. In our case, every point on the line has the y-coordinate of 5. Hence, the line \(y = 5\) is parallel to the x-axis and follows the property of parallel lines as it shares the same slope of 0.