Problem 46 - College Algebra

Question

For the following exercises, find the \(x\) - and \(y\) -intercepts of each equation. $$ 7 x+2 y=56 $$

Step-by-Step Solution

Answer

x-intercept: (8, 0); y-intercept: (0, 28).

Step 1: Finding the x-intercept

To find the x-intercept of the equation, we set the y-value to zero. This gives us the equation: \[ 7x + 2(0) = 56 \] Simplifying, we obtain: \[ 7x = 56 \] To solve for \( x \), divide both sides by 7: \[ x = \frac{56}{7} = 8 \] Therefore, the x-intercept is \((8, 0)\).

Step 2: Finding the y-intercept

To find the y-intercept, we set the x-value to zero. This gives us the equation: \[ 7(0) + 2y = 56 \] Simplifying this, we have: \[ 2y = 56 \] To solve for \( y \), divide both sides by 2: \[ y = \frac{56}{2} = 28 \] Thus, the y-intercept is \((0, 28)\).

Key Concepts

Understanding Linear Equations

A linear equation is an equation that forms a straight line when plotted on a coordinate plane. It is composed of variables and constants, with the highest degree of the variables being one. For example, in the equation \(7x + 2y = 56\), both \(x\) and \(y\) are to the power of one, making it linear. Linear equations are fundamental in algebra because they help us understand relationships between variables. They are straightforward to work with, as the solutions tend to be unique or infinite and are easy to interpret. Remembering the basic form of a linear equation, which is \(y = mx + b\), can help identify the slope \(m\) and the y-intercept \(b\) quickly. However, equations may appear in different forms, like the standard form \(Ax + By = C\), which is what we have here. Mastering linear equations sets the foundation for more complex algebraic concepts.

Solving for Variables in Algebra

Solving for variables in an equation involves finding the numerical value of unknowns that make the equation true. In algebra, variables are symbols like \(x\), \(y\), or more that stand in for unknown numbers. The process of solving involves manipulating the equation by using different operations to isolate the variable you are solving for.

• Subtract or add terms to both sides to simplify the equation.
• Multiply or divide both sides to isolate the variable.
• Check the solution by plugging it back into the original equation.

In our exercise, we solve for \(x\) and \(y\) by setting the other variable to zero, isolating it using arithmetic operations. This method of substitution is crucial for finding intercepts when dealing with linear equations. Grasping this concept allows for solving various algebra problems efficiently.

Intercepts in Algebra

Intercepts are points where a line crosses the axes on a graph. There are two main types: the x-intercept and the y-intercept. Each provides crucial information about the graph of an equation.

• The x-intercept is where the line crosses the x-axis, meaning the y value at this point is zero.
• The y-intercept is where the line crosses the y-axis, where the x value is zero.

To find the x-intercept in our equation \(7x + 2y = 56\), we substitute \(y = 0\) and solve for \(x\), resulting in the point \((8, 0)\). Similarly, to find the y-intercept, substitute \(x = 0\) and solve for \(y\), yielding the point \((0, 28)\). These intercepts help us understand not only where a line is positioned on a graph but also provide points through which the line definitely passes, making them essential in graphing and interpreting linear equations.