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Chapter 8: Problem 120

Solve by graphing. $$ x-1=-5 $$

Short Answer

Expert verified
\( x = -4 \).

Step by step solution

01

Identify the equation

The given equation is \( x - 1 = -5 \).
02

Solve for x

Add 1 to both sides of the equation to isolate \( x \).\( x - 1 + 1 = -5 + 1 \)This simplifies to \( x = -4 \).
03

Graph the equation

Plot the point where \( x = -4 \) on the x-axis. This is a vertical line crossing the x-axis at \( -4 \). Locate \( -4 \) on a number line and draw a point there.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Equations
Solving equations is a foundational skill in algebra that requires finding the value of the variable that makes the equation true. In our original exercise, we are given the equation:

\[ x - 1 = -5 \].
To solve this, we need to isolate the variable on one side of the equation. The steps involve performing arithmetic operations to both sides to maintain equality. To solve for \( x \), we identify and execute the following step:

  • Add 1 to both sides to cancel out the -1 on the left-hand side:
  • \[ x - 1 + 1 = -5 + 1 \] which simplifies to \[ x = -4 \].

Now, you have found that \( x = -4 \) is the solution that balances the original equation.
Graphing Linear Equations
Graphing linear equations is a way to visually represent solutions on the coordinate plane. For the equation

\[x - 1 = -5\],

we have determined from the previous steps that the value of \(x\) is -4. To graph this:

  • Locate -4 on the x-axis.
  • Draw a point at -4 and it represents the solution \(x = -4\).
For this specific type of equation, it will appear as a vertical line crossing through the x-axis at -4 since the solution for \(x\) is constant regardless of the value of \(y\). This helps in visualizing that all points on this line share the same x-coordinate (-4).
Isolation of Variables
Isolation of variables is a crucial step in solving algebraic equations, where you manipulate the equation to have the variable of interest alone on one side. In the problem \(x - 1 = -5\), to isolate \(x\), you perform the following steps:

  • Identify the operation that is affecting the variable (here it is -1).
  • Apply the inverse operation to both sides to cancel out the effect (adding 1 to both sides in this case).
  • \[ x - 1 + 1 = -5 + 1 \] simplifies to \[ x = -4 \]

This method ensures the variable is isolated, making it easier to determine its value. When isolating variables, always perform the same operation on both sides to keep the equation balanced and do the opposite of what is currently applied to the variable (addition cancels subtraction, and vice versa). This consistent approach can be applied to solve more complex algebraic equations as well.

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