Problem 22 - Algebra 1: Concepts and Skills

Question

Find the x-intercept of the line. $$ -6 x-4 y=42 $$

Step-by-Step Solution

Answer

The x-intercept of the line \(-6x - 4y = 42\) is at the point \((-7, 0)\)

Step 1: Identify the y-term and set it to zero

The equation provided is \(-6x - 4y = 42\). The y-term here is '-4y'. The x-intercept is when y=0, so set '-4y' as 0 which leaves \(-6x = 42\).

Step 2: Solve for x

Solve the equation \(-6x = 42\) for x. This is done by dividing both sides of the equal sign by coefficient of x, which is -6. The resulting x value is \(x = -7\).

Step 3: Present the x-intercept

The x-intercept is at the point where x = -7. In coordinate form, this would be presented as \((-7, 0)\).

Key Concepts

Linear Equations

Linear equations are mathematical expressions that establish a straight line when graphed on a coordinate plane. An equation is considered linear if it can be expressed in the standard form of a straight line, like \(ax + by + c = 0\). In these equations, the highest power of the variables is always one.

Some key characteristics of linear equations include:

• They have up to two variables, often denoted by \(x\) and \(y\).
• The graph of a linear equation is a straight line.
  
• Every solution to the equation corresponds to a point on this line.
  

Understanding linear equations is crucial because they are often foundational to understanding more complex algebraic concepts.

In our original exercise, the linear equation \(-6x - 4y = 42\) needs to be rearranged or manipulated to find specific points, such as intercepts.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This branch of mathematics allows us to represent geometric figures, such as lines and curves, through algebra.

A coordinate plane consists of two perpendicular lines, called axes, which intersect at the origin (0,0). The horizontal line is called the x-axis and the vertical is the y-axis. Together, these axes form a grid on which we can plot points, lines, and curves.

The x-intercept is a specific application of coordinate geometry. It is the point where a line or curve crosses the x-axis. At this point, the value of \(y\) is zero. In our exercise, after setting \(y = 0\) in the equation \(-6x - 4y = 42\), we found that the x-intercept is at \((-7, 0)\). This means the line crosses the x-axis at x = -7.

Understanding how to find intercepts is a vital skill in coordinate geometry as it helps in graphing lines and understanding their behavior on the plane.

Solving Equations

Solving equations involves finding the value(s) of the variable(s) that make the equation true. This process often involves several arithmetic operations such as addition, subtraction, multiplication, or division.

In the context of our exercise, solving the equation means finding the x-intercept. The given equation is \(-6x - 4y = 42\). To find the x-intercept, we set \(y\) to zero and solved the resulting equation for \(x\).

Here's how this works step-by-step:

• Identify the relevant term. In this case, set \(y = 0\), transforming the equation to \(-6x = 42\).
  
• Isolate \(x\) by dividing both sides by -6 to get \(x = -7\).
• Thus, the x-intercept is \((-7, 0)\).
  

Solving equations is a fundamental skill in algebra, critical for accurately finding solutions to mathematical problems and real-world scenarios alike.