Question

Find the \(x\) -intercept of the graph of the equation. $$ y=x-\frac{1}{4} $$

Short answer

The x-intercept of the graph is \(x=\frac{1}{4}\)

Step-by-step solution

Set y to zero

In order to find the x-intercept, it is important to set the value of 'y' to zero because at the x-axis intercept, the value of y is zero. Therefore, we adjust our equation to reflect this change: \( 0=x-\frac{1}{4}\).

Solve for x

With our modified equation, now it's time to solve for 'x'. Starting with \(0=x-\frac{1}{4}\), we need to isolate 'x' on one side of the equation. To do this, add \(\frac{1}{4}\) to both sides of the equation. This results in an equation of \(x=\frac{1}{4}\).

Key concepts

Solving Equations

Understanding how to solve equations is a foundational skill in algebra. When we talk about 'solving equations', we're looking for the variable's value that makes the equation true. For instance, finding the x-intercept of a line involves solving the equation for when the value of y is zero.

The process usually involves isolating the variable on one side of the equation using arithmetic operations such as addition, subtraction, multiplication, and division. For the given exercise where the equation is \( y=x-\frac{1}{4} \), setting \( y=0 \) and solving for \( x \) involves simply adding \( \frac{1}{4} \) to both sides. This isolation of \( x \) gives us the exact point where the line crosses the x-axis, which is an essential aspect of understanding how graphs behave in a coordinate system.

Why Solve for x?

When we set the value of y to zero, we effectively find the point where our line will intersect the x-axis. This is the practical reason behind why we solve for x in this context.

Graphing Linear Equations

Graphing linear equations is another integral part of algebra, allowing us to visualize solutions to equations on a coordinate plane. Every linear equation can be represented by a straight line, and each line is defined by its slope and intercepts. To graph a linear equation like \( y=x-\frac{1}{4} \), we find the intercepts and then use the slope to determine other points on the line.

By plotting the x-intercept obtained after solving \( 0=x-\frac{1}{4} \) which gives us \( x=\frac{1}{4} \), we can start to see the graph take shape. Typically, you would also find the y-intercept where \( x=0 \) and plot that point. From there, you can draw a line through both intercepts to get the full graph of the equation.

Clarity in Visual Representation

Always remember that graphing not only provides a visual representation of the equation but also offers insight into the relationship between the variables involved. By identifying the point where the line crosses the x-axis, you're reinforcing your understanding of how x and y interact in an equation.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations. In our exercise, \( x-\frac{1}{4} \) is an example of an algebraic expression that represents a line when paired with \( y= \). These expressions become particularly useful when we want to describe relationships between variables or define the rules of geometrical shapes.

An expression becomes an equation when we include an equals sign, as with a linear equation. Then, solving that equation is what allows us to find specific points that satisfy the equation, such as intercepts on a graph. The transition from algebraic expression to solving the equation is a critical journey in algebra which combines comprehension of expressions and their resolution.

Building Blocks of Equations

The manipulation of these expressions is key to reshaping equations and ultimately solving them. Whether combining like terms, factoring, or expanding expressions, each of these skills leads to a deeper understanding of how equations behave and how we can find their solutions.