Question

Use the equation \(2 x+7 y=14\). What is the \(x\) -intercept?

Short answer

The x-intercept of the line represented by the equation \(2x + 7y = 14\) is 7.

Step-by-step solution

Set y equal to zero

As the point of interest is the x-intercept, where the line intersects the x-axis, the value of y at this point would be zero. Hence, for the given equation \(2x + 7y = 14\), replace y with 0 to get \(2x + 7(0) = 14\). The equation simplifies to \(2x = 14\).

Solve for x

After replacing the value of y, solve for x by dividing both sides of the equation by 2. This gives \(x = 14/2 = 7\).

Key concepts

Understanding Linear Equations

In algebra, a linear equation is a statement of equality between two algebraic expressions in which the highest power of the variable is one. Simply put, it's an equation of the first degree. The general form of a linear equation in two variables, x and y, can be expressed as \( ax + by = c \), where \( a \) and \( b \) are coefficients, and \( c \) is the constant.

When dealing with a linear equation, finding the x-intercept is a common task. The x-intercept is the point at which the graph of the equation crosses the x-axis. By definition, the y-coordinate of any point on the x-axis is zero. To find the x-intercept, you substitute zero for y in the equation and solve for x, which gives you the point where the line touches the x-axis.

For example, using the equation from the exercise \(2x + 7y = 14\), we plug in zero for y and solve for x, leading to an x-intercept at (7, 0).

Graphing Lines

To graph a line represented by a linear equation, you typically need two points through which the line passes. One way to find these points is by determining the intercepts. We've already found one point by calculating the x-intercept. To find the y-intercept, where the line crosses the y-axis, we would set x to zero and solve for y.

After identifying the intercepts, plot these points on a coordinate plane. Then, draw a line through the plotted points, extending it in both directions. This visual representation helps understand the behavior of the linear equation in a graphical form. If you're given a linear equation like \(2x + 7y = 14\), you can graph the line by first finding the intercepts, which serve as your guide points.

Solving Algebraic Equations

The process of solving algebraic equations involves finding the values for the variables that make the equation true. Solving a linear equation, as one of the simplest algebraic equations, usually entails a series of operations that will isolate the variable on one side of the equation.

Following the steps from the given exercise, the first operation involves replacing y with zero to focus on finding the x-value of the x-intercept. Next, you simplifying the equation by removing any terms that have been neutralized by the substitution (in this case, 7y becomes 0). Finally, applying basic arithmetic operations—such as division, in this example—gives us the solution for x.