Problem 19 - Algebra 1

Question

Find the \(x\) -intercept of the graph of the equation. $$ 2 x+2 y=-10 $$

Step-by-Step Solution

Answer

The x-intercept of the graph of the equation is (-5, 0).

Step 1: Setting y equal to 0

Substitute 0 for y in the equation 2x + 2y = -10, we get 2x + 2*0 = -10, which simplifies to 2x = -10.

Step 2: Solve for x

To isolate x, divide both sides of the equation by 2, we get x = -10/2, which simplifies to x = -5.

Step 3: Write the x-intercept

The x-intercept is a point, so it is written as an ordered pair, consequently should be written as (-5, 0).

Key Concepts

Solving Linear Equations

Solving linear equations is a fundamental skill in algebra. Linear equations are equations of the first degree, meaning the highest power of the variable is one. In the given exercise, the equation is \(2x + 2y = -10\). Here are some key points to remember when solving such equations:

• Identify and isolate the variable: Choose the variable you want to solve for first. In this case, to find the x-intercept, you need to solve for \(x\) by setting \(y\) to zero.
• Simplify the equation: Substitute any given values (like \(y = 0\)) into the equation and simplify step-by-step. This involves performing operations like addition, subtraction, multiplication, or division.
• Check your solution: After isolating \(x\) so that you have \(x = \frac{-10}{2}\), simplify to find \(x\), which gives \(x = -5\).

Learning to solve linear equations accurately and efficiently helps in understanding more complex algebraic concepts later.

Graphing Equations

Graphing equations allows you to visually represent mathematical relationships on the coordinate plane. To graph the equation \(2x + 2y = -10\), it's important to find the intercepts:

• The x-intercept is found by setting \(y = 0\) and solving for \(x\). In this scenario, that intercept is the point \((-5, 0)\).
• The y-intercept, conversely, is found by setting \(x = 0\) and solving for \(y\). While not directly required in this task, finding the y-intercept provides a clearer understanding of the graph's orientation.

With the intercepts determined, you can plot them on a graph. Draw a straight line through these points since a linear equation graphs as a straight line. This visual representation aids not only in understanding the solution but also in seeing how the equation behaves in a geometrical space.

Coordinate Plane

The coordinate plane is a two-dimensional plane defined by a horizontal axis (x-axis) and a vertical axis (y-axis). Each point on this plane is represented by an ordered pair \((x, y)\). For linear equations, visualizing solutions on this plane provides a clear understanding of the relationship within the equation.

• The x-axis is the line where \(y\) is zero. Points on this line have coordinates of the form \((x, 0)\), just like the x-intercept \((-5, 0)\) from the example.
• The y-axis, on the other hand, is where \(x\) is zero. Points here have coordinates \((0, y)\).

Understanding how to read and plot points on the coordinate plane is crucial not only for graphing lines but for making sense of many real-world situations where relationships between two quantities need to be represented geometrically. The coordinate plane provides a clear way to visualize linear equations and their intercepts.