Problem 20 - Algebra 1

Question

Find the \(x\) -intercept of the graph of the equation. $$ 3 x+4 y=12 $$

Step-by-Step Solution

Answer

The x-intercept of the graph of the equation is \(x = 4\).

Step 1: Set \(y=0\)

The x-intercept is the value of \(x\) when \(y = 0\). Therefore, set \(y = 0\) in the equation, which gives \(3x + 4(0) = 12\). This simplifies to \(3x = 12\).

Step 2: Solve for \(x\)

Next, isolate \(x\) to find the x-intercept by dividing both sides by 3. This gives \(x = 12 / 3\), which further simplifies to \(x = 4\).

Key Concepts

Linear Equations

Linear equations are equations of the first degree, meaning they have no variables raised to a power higher than one. A standard form for a linear equation in two variables, like the one in our exercise, is given as \(Ax + By = C\). Here, \(A\), \(B\), and \(C\) are constants and \(x\) and \(y\) are variables. The beauty of linear equations lies in their straightforwardness, and the solutions typically represent a line when plotted on a graph.

When tackling these equations, you often work to find specific points such as intercepts. The intercepts are where the line crosses the x-axis and y-axis. To find these points, you manipulate the equation by setting either \(x\) or \(y\) to zero, allowing us to solve for the other variable. This leads to defining important features of the graph.

Graphing

Graphing linear equations involves plotting the line described by the equation on a coordinate plane. The line showcases all possible solutions \((x, y)\) of the equation and typically forms a straight line because of the equation's linear nature.

One of the simplest ways to start graphing a line is to determine its intercepts. For example, the x-intercept is found by setting \(y = 0\) in the equation and solving for \(x\). Similarly, the y-intercept is located by setting \(x = 0\) and solving for \(y\). Once these points are plotted, you can draw a line through them.

Graphing provides a visual representation that helps in understanding abstract algebraic equations. It allows us to see the relationship between the variables and offers a practical way of finding solutions by checking where the line lies in relation to other graph components, like axes and other lines.

Solving Equations

Solving equations is a key mathematical skill that involves finding unknown variables. In the case of linear equations, we isolate the desired variable through a series of algebraic steps.

Consider the equation in our exercise: \(3x + 4y = 12\). To find the x-intercept, we set \(y = 0\) and simplify:

• Substitute 0 for \(y\): \(3x + 4(0) = 12\)
• This simplifies to \(3x = 12\)
• Divide both sides by 3 to solve for \(x\): \(x = 4\)

By isolating \(x\), we determined that the x-intercept is at the point (4, 0) on the graph.

This process involves balancing the equation with careful manipulation of terms until the variable of interest is isolated on one side of the equation. Understanding how to methodically solve equations reveals the logical structure behind the math, aiding in clear comprehension.