Problem 45 - Elementary Algebra

Question

For Problems \(45-60\), write the equation of the line that satisfies the given conditions. Express final equations in standard form. \(x\) intercept of 2 and \(y\) intercept of \(-4\)

Step-by-Step Solution

Answer

The equation of the line is \(2x - y = 4\).

Step 1: Understand the Intercepts

We have an x-intercept of 2 and a y-intercept of -4. This means the line crosses the x-axis at the point (2,0) and the y-axis at the point (0,-4).

Step 2: Use the Intercept Form Formula

The intercept form of the equation of a line is \( \frac{x}{a} + \frac{y}{b} = 1 \), where \(a\) is the x-intercept and \(b\) is the y-intercept. We substitute \(a = 2\) and \(b = -4\) into the formula.

Step 3: Substitute Intercepts into the Formula

Using \( \frac{x}{2} + \frac{y}{-4} = 1 \), we have the equation \( \frac{x}{2} - \frac{y}{4} = 1 \).

Step 4: Eliminate Fractions

To eliminate the fractions, multiply every term by the least common multiple of 2 and 4, which is 4: \(4\left( \frac{x}{2} \right) - 4\left( \frac{y}{4} \right) = 4\cdot 1\), leading to \(2x - y = 4\).

Step 5: Write the Final Equation in Standard Form

The equation \(2x - y = 4\) is already in standard form, which is \(Ax + By = C\).

Key Concepts

X-intercept

The x-intercept of a line is the point where the line crosses the x-axis. This means the value of y is zero at this point. For instance, in our problem, the x-intercept is 2, so the line touches the x-axis at the point (2, 0). Understanding the x-intercept is crucial because it provides where the line interacts with the x-axis.

• The x-intercept is found by setting y = 0 in the equation of the line and solving for x.
• This is a helpful starting point when analyzing or graphing linear equations, especially when finding and plotting the line on a graph.

By recognizing this, you can quickly identify one significant feature of the line on the coordinate plane, helping in visualizing and understanding the line's behavior.

Y-intercept

The y-intercept of a line is the point where the line crosses the y-axis. Here, the value of x is zero. In our particular situation, the y-intercept is -4, indicating the line meets the y-axis at (0, -4). Grasping the y-intercept concept is essential for interpreting the line's relationship with the y-axis.

• The y-intercept occurs when you set x = 0 in your equation and solve for y.
• This point is particularly important in graphing lines from equations, as it is often easy to spot and use for plotting.

The y-intercept gives additional context to a line's placement as it travels across the graph, revealing its intersection with the y-axis.

Standard Form

The standard form of a linear equation is written as \(Ax + By = C\), where A, B, and C are integers, and A is typically positive. In our solved problem, the line's equation is in the form of \(2x - y = 4\).

• Standard form is particularly useful for signaling linear relationships in a straightforward manner.
• It is often used for efficiently describing vertical and horizontal lines.

Standard form simplifies understanding the balance between x and y values that satisfy the equation, making it easier to analyze and evaluate the equation without complex algebraic manipulations.

Algebra Problems

Solving algebra problems often involves rearranging equations, understanding different forms of linear equations, and knowing how these equations appear on a graph. In our example, we worked through several algebraic concepts:

• Substituting given values into a formula to make sense of intercepts.
• Manipulating an expression to convert it into standard form by eliminating fractions, such as multiplying through by the least common multiple.

Effective problem-solving in algebra involves recognizing patterns, thinking critically about the given information, and making logical deductions—a skill set extremely valuable in both academic and real-world situations.