Problem 29 - Understanding Elementary Algebra with Geometry

Question

Write an equation of the line satisfying the given conditions. Line has \(x\) -intercept 5 and \(y\) -intercept 2

Step-by-Step Solution

Answer

\( \frac{x}{5} + \frac{y}{2} = 1 \) or \( 2x + 5y = 10 \)

Step 1: Understand the intercepts

The line has an x-intercept of 5, which means it crosses the x-axis at the point (5, 0). Similarly, the line has a y-intercept of 2, meaning it crosses the y-axis at the point (0, 2).

Step 2: Write down the general form of the intercepts

The x-intercept is (5, 0) and the y-intercept is (0, 2). These points can be used in the intercept form of the equation of a line: \[ \frac{x}{a} + \frac{y}{b} = 1 \] where \( a \) and \( b \) are the x-intercept and y-intercept, respectively.

Step 3: Substitute the intercepts into the equation

Substituting \( a = 5 \) and \( b = 2 \) into the intercept form equation: \[ \frac{x}{5} + \frac{y}{2} = 1 \]

Step 4: Convert to standard linear form (if desired)

To convert the equation to the standard linear form (Ax + By = C), multiply through by the least common multiple of the denominators (10): \[ 10 \left( \frac{x}{5} + \frac{y}{2} \right) = 10 \times 1 \] \[ 2x + 5y = 10 \]

Key Concepts

x-intercept

The x-intercept is where the line touches the x-axis. At this point, the value of y is always 0. For example, if a line has an x-intercept of 5, it means the line crosses the x-axis at the point (5, 0). This point tells us that if we set y to 0 in the line's equation, solving for x will give us 5.

This point is essential for finding the equation of the line. When you have the x-intercept and another point or the slope, you can determine the entire line. Just remember, for the x-intercept:

• The y-coordinate is always 0.
• Set y to 0 in the equation to find the x-intercept.

y-intercept

The y-intercept is where the line touches the y-axis. Here, the value of x is always 0. For a line with a y-intercept of 2, it crosses the y-axis at the point (0, 2). This point is crucial for understanding the vertical placement of the line on the graph. If you set x to 0 in the line's equation, solving for y will give you the y-intercept.

Knowing the y-intercept helps to quickly sketch or understand the line's position. Key points to keep in mind:

• The x-coordinate is always 0 at this point.
• Set x to 0 in the equation to find the y-intercept.

linear equations

Linear equations describe lines in a coordinate plane. They typically follow the form: \ 'y = mx + b' \ where 'm' is the slope and 'b' is the y-intercept. However, there are multiple ways to write these equations.

Another form is the intercept form, especially useful when you know the x-intercept (a) and y-intercept (b): \ \( \frac{x}{a} + \frac{y}{b} = 1 \) \ For example, if the x-intercept is 5 and the y-intercept is 2, substituting these values into the equation gives: \ \( \frac{x}{5} + \frac{y}{2} = 1 \) \
Sometimes, it may be necessary to convert this into the standard form (Ax + By = C):\

• Multiply through by the least common multiple of the denominators to clear fractions.
• Simplify to get the linear equation in standard form.

Understanding these forms and when to use them can make working with linear equations simpler.