X = a

Definition

The expression 'x = a' represents an equation where the variable 'x' is equal to a specific constant value 'a'. This term is particularly important in the context of graphing linear equations in two variables, as it helps define the relationship between the variables and the resulting graph.

5 Must Know Facts For Your Next Test

1. When $x = a$, the linear equation $y = mx + b$ simplifies to $y = mb + b$, which represents a horizontal line with a $y$-intercept of $b$.
2. The graph of the equation $x = a$ is a vertical line that passes through the point $(a, 0)$ on the coordinate plane.
3. The expression $x = a$ is often used to represent a constraint or a condition in a system of linear equations, where the value of $x$ is fixed.
4. Solving systems of linear equations with the condition $x = a$ can be done by substituting the value of $a$ into the other equation(s) and solving for the remaining variable(s).
5. The equation $x = a$ is commonly used in the context of linear programming, where it represents a constraint on the feasible region of the optimization problem.

Review Questions

• Explain how the equation $x = a$ simplifies the graph of a linear equation in the form $y = mx + b$.
  • When $x = a$, the linear equation $y = mx + b$ simplifies to $y = mb + b$, which represents a horizontal line with a $y$-intercept of $b$. This is because the value of $x$ is fixed at $a$, and the equation becomes a function of $y$ alone. The resulting graph is a vertical line passing through the point $(a, 0)$ on the coordinate plane.
• Describe how the equation $x = a$ is used in the context of solving systems of linear equations.
  • The expression $x = a$ is often used to represent a constraint or a condition in a system of linear equations. When solving such a system, the value of $a$ can be substituted into the other equation(s) to simplify the problem. This allows the remaining variable(s) to be solved for, as the system is reduced to a single equation in one unknown. Solving systems of linear equations with the condition $x = a$ is a common technique in linear programming and other applications where specific constraints are imposed on the variables.
• Analyze the role of the equation $x = a$ in the context of linear programming and optimization problems.
  • In the field of linear programming, the equation $x = a$ is used to represent a constraint on the feasible region of the optimization problem. By fixing the value of $x$ to a specific constant $a$, the problem is simplified, and the optimization can be performed with one less variable to consider. This constraint can be combined with other linear equations and inequalities to define the feasible region, which is the set of all possible solutions that satisfy the given conditions. The equation $x = a$ plays a crucial role in linear programming by allowing for more efficient and targeted optimization of the objective function within the defined constraints.

Related terms

Linear Equation:

A linear equation is an equation that can be written in the form $ax + by + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ and $y$ are the variables.

Slope-Intercept Form:

The slope-intercept form of a linear equation is $y = mx + b$, where $m$ represents the slope and $b$ represents the $y$-intercept.

Point-Slope Form:

The point-slope form of a linear equation is $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a known point on the line and $m$ is the slope.