Problem 4 - Algebra 1

Question

Find the \(x\) -intercept of the graph of the equation. $$ y=2 x+20 $$

Step-by-Step Solution

Answer

The x-intercept of the given graph is \(x=-10\).

Step 1: Set y equal to zero

The x-intercept is where the graph touches the x-axis, which is when 'y' is equal to zero. So set \(y=0\) in the equation.

Step 2: Solving for x

The equation now becomes \(0=2x+20\). Now, we need to solve this equation for 'x'. Subtract 20 from both sides of the equation to isolate the term with 'x', giving \(-20=2x\).

Step 3: Isolate x

In order to isolate 'x', divide by 2 across the equation. This gives \(x=-20/2\).

Step 4: Simplify the Expression

Simplify the expression, so it results in \(x=-10\). This implies that the graph crosses the x-axis at the point where x=-10.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that form a straight line when plotted on a graph. They are called "linear" because their graph is a straight line. These equations typically have two variables, often represented as "x" and "y". These variables represent points on the x and y axes of a coordinate plane, respectively. A standard form for a linear equation is given by \(ax + by = c\), where "a", "b", and "c" are constants.
Linear equations can be written in several forms:

• Standard Form: \(ax + by = c\)
• Slope-Intercept Form: \(y = mx + b\)
• Point-Slope Form: \(y - y_1 = m(x - x_1)\)

In these equations, the coefficient "m" represents the slope, which shows how steep the line is. The constant "b" (in slope-intercept form) is the y-intercept, the point where the line crosses the y-axis. Linear equations are fundamental because they model real-world situations where relationships between two variables are proportional and predictable.

Coordinate Plane

The coordinate plane is a two-dimensional surface formed by two perpendicular number lines intersecting at a point called the origin. These number lines are referred to as the x-axis (horizontal) and the y-axis (vertical). The resulting grid allows you to graph equations and easily see relationships between numbers.
Each point on the coordinate plane is represented by an ordered pair \((x, y)\), where "x" is the distance from the y-axis and "y" is the distance from the x-axis. This ordered pair is crucial for graphing linear equations and finding intercepts. For example, the x-intercept occurs when \(y=0\), and the y-intercept happens when \(x=0\).
The coordinate plane is divided into four quadrants:

• Quadrant I: both x and y are positive
• Quadrant II: x is negative and y is positive
• Quadrant III: both x and y are negative
• Quadrant IV: x is positive and y is negative

Understanding the coordinate plane allows you to graph linear equations accurately and recognize patterns or trends present in the data.

Solving Equations

Solving equations means finding the value(s) of the variable(s) that make the equation true. When solving, we perform operations to isolate the variables step by step.
In the context of finding the x-intercept of a linear equation, we follow a systematic method:

• Step 1: Set \(y\) to zero. This step helps find where the equation touches the x-axis.
• Step 2: Simplify the equation. With \(y\) set to zero, simplify terms to isolate the variable "x".
• Step 3: Use basic algebraic operations. You may need to add, subtract, multiply, or divide to solve for "x".

For our example, solving \(y = 2x + 20\) with \(y = 0\) involves subtracting 20 to get \(-20 = 2x\), then dividing by 2 to isolate \(x\). This results in \(x = -10\), so the x-intercept is \(-10\).
Solving equations effectively requires a solid understanding of algebraic principles and is essential for tackling more complex mathematical problems.