Problem 24 - Algebra 1

Question

Find the \(x\) -intercept of the graph of the equation. $$ 2 x+6 y=-24 $$

Step-by-Step Solution

Answer

The x-intercept of the given equation is \(-12\).

Step 1: Set y to Zero

Setting y to zero in the equation \(2x + 6y = -24\) gives \(2x+6*(0) = -24\). This simplifies to \(2x = -24\).

Step 2: Solve For x

To isolate x, divide both sides of the equation by 2. This gives \(x = -24 / 2\).

Step 3: Find the Value of x

Solving the equation should give \(x = -12\). The coordinates of the x-intercept are thus \((-12, 0)\).

Key Concepts

Linear Equations

Linear equations are equations that make a straight line when graphed on a coordinate plane. They are commonly written in the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. In our exercise, the equation is \(2x + 6y = -24\). This is a linear equation because it involves only the variables \(x\) and \(y\) raised to the first power, without any curves or higher-degree terms.

To identify a linear equation, look for both variables being in the first degree and no multiplication between them or squaring. Such equations will produce a straight line whenever you graph them. Linear equations are foundational in algebra and help solve numerous practical problems.

They can tell us about relationships between two quantities, helping us predict values and understand inherent trends. They're quite versatile, appearing in various fields like engineering, economics, and natural sciences.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can graphically represent equations. It's made up of two number lines: the horizontal axis, called the \(x\)-axis, and the vertical axis, called the \(y\)-axis. The point where these axes intersect is called the origin, labeled as \((0,0)\).

When we plot points on this plane, each point's location is determined by an \(x\)-coordinate and a \(y\)-coordinate. Returning to the exercise's x-intercept of \((-12, 0)\), it is where the graph of the equation \(2x + 6y = -24\) crosses the \(x\)-axis. It's important to note at the x-intercept, the value of \(y\) is always zero.

Understanding how to use and interpret the coordinate plane is crucial. It allows you to visualize equations and solutions, making abstract algebraic concepts more tangible. You can better understand distance, slope, and intersection points with practice and familiarity.

Solving Equations

Solving equations involves finding the value of the unknown variable that makes the equation true. In this exercise, we dealt with finding the \(x\)-intercept by solving the linear equation \(2x + 6y = -24\).

• Step 1: Set \(y\) to zero since at the \(x\)-intercept, \(y\) is always zero. You are left with \(2x = -24\).
• Step 2: Solve for \(x\) by dividing both sides of the equation by 2, yielding \(x = -12\).

Hence, the \(x\)-intercept is the point \((-12, 0)\) on the coordinate plane. This process of solving involves a few common techniques:

• Isolating the variable you're solving for.
• Performing the same operation on both sides of the equation to maintain equality.
• Simplifying the equation to get the variable by itself.

These methods are versatile and apply to various equations beyond linear ones, making them a core algebraic skill.