Problem 46 - Algebra 1

Question

Graph the equation. $$2 x-4 y=24$$

Step-by-Step Solution

Answer

The x-intercept of the graph is (12, 0) and the y-intercept is (0, -6). The line passing through these points represents the graph of the equation \(2x - 4y = 24\).

Step 1: Find the X-intercept

Set y = 0 in the equation and solve for x. So, the equation becomes \(2x - 4 \cdot 0 = 24\), which simplifies to \(2x = 24\). Solving for x results x = 12. So, the x-intercept is (12, 0).

Step 2: Find the Y-intercept

Set x = 0 in the equation and solve for y. So, the equation becomes \(2 \cdot 0 - 4y = 24\), which simplifies to \(-4y = 24\). Solving for y results y = -6. So, the y-intercept is (0, -6).

Step 3: Plot the points and draw the line

Plot the points (12, 0) and (0, -6) on the coordinate plane, then draw a line passing through these points. The line represents the graph of the equation \(2x - 4y = 24\).

Key Concepts

X-intercept

In the context of graphing linear equations, the x-intercept is a key concept representing the point where the graph of a line crosses the x-axis. To determine the x-intercept from an equation, set the y-value to zero and solve for x.
For the equation \(2x - 4y = 24\), you find the x-intercept by substituting \(y=0\), leading to the equation \(2x = 24\). Solving gives \(x = 12\). Thus, the x-intercept is the point \((12, 0)\) on the coordinate plane.
Remember that the x-intercept always has zero as its y-coordinate because this is where the line touches the x-axis without rising or falling above or below it.

Y-intercept

The y-intercept is similarly important in graphing lines, as it shows where the line crosses the y-axis. To find the y-intercept, set the x-value to zero within the equation.
For \(2x - 4y = 24\), substitute \(x=0\) to get \(-4y = 24\). Solving this equation, we find \(y = -6\). Hence, the y-intercept is the point \((0, -6)\).
This point always has zero as its x-coordinate because it represents where the line intersects the y-axis, standing purely at the vertical level without moving horizontally.

Coordinate Plane

The coordinate plane, often essential in graphing, is a two-dimensional surface formed by two perpendicular number lines: the x-axis and the y-axis. These axes intersect at a point called the origin, which has coordinates \((0, 0)\).

• The x-axis is the horizontal line that increases to the right and decreases to the left.
• The y-axis is the vertical line that increases upwards and decreases downwards.

Points plotted on the coordinate plane are written as ordered pairs \((x, y)\), where 'x' represents the horizontal position and 'y' the vertical position.
Graphing lines involves plotting points, such as the x- and y-intercepts, and connecting them with a straight line to represent the equation.

Equation of a Line

An equation of a line in two variables, like \(2x - 4y = 24\), represents all the points on a straight line in the coordinate plane.
This particular form is known as the standard form of a linear equation \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants.

• Linear equations can also appear in slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) the y-intercept.

Each line equation describes a unique flat surface in the plane, characterized by consistently rising or falling (determined by its slope) and specific intersections with the axes (their intercepts).
Translating an equation into a graph requires finding values like intercepts and plotting those, which act as visual proof of the mathematical relationship expressed by the equation.