The constant term is a numerical value in an equation that does not change, regardless of the values assigned to the variables. It is a fixed quantity that is independent of the variables in the equation.
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The constant term in a linear equation does not change when the values of the variables are changed.
The constant term represents the y-intercept of the graph of the linear equation.
In a system of linear equations, the constant terms are used to determine the solution or set of solutions.
The constant term is often denoted by the letter 'b' in the standard form of a linear equation: $ax + by = c$.
The value of the constant term can affect the behavior of the linear equation, such as the direction of the line and the location of the y-intercept.
Review Questions
Explain the role of the constant term in the standard form of a linear equation, $ax + by = c$.
In the standard form of a linear equation, $ax + by = c$, the constant term $c$ represents the y-intercept of the graph of the equation. The constant term is a fixed value that does not change, regardless of the values assigned to the variables $x$ and $y$. The constant term is an essential component of the equation, as it determines the vertical position of the line on the coordinate plane.
Describe how the constant term affects the behavior of a linear equation in a system of linear equations.
In a system of linear equations, the constant terms are used to determine the solution or set of solutions. The constant terms, along with the coefficients of the variables, define the specific lines in the system. The values of the constant terms can affect the behavior of the system, such as whether the lines intersect, are parallel, or coincide. The constant terms play a crucial role in identifying the unique solution, if it exists, or determining the type of solution set for the system of linear equations.
Analyze the relationship between the constant term and the y-intercept of the graph of a linear equation.
The constant term in a linear equation is directly related to the y-intercept of the graph of the equation. The y-intercept is the point where the line crosses the y-axis, and its value is equal to the constant term. This relationship is evident in the standard form of a linear equation, $ax + by = c$, where the constant term $c$ represents the y-intercept. Understanding the connection between the constant term and the y-intercept is essential for interpreting the graphical representation of a linear equation and solving problems involving systems of linear equations.