3 Vocabulary Terms - Define/Illustrate
1) Intersection point of two lines
2) System of linear equations
3) Solution of a system as an ordered pair
1) Intersection point of two lines: The intersection point of two lines is the point where the two lines intersect or meet. It is the location where the two lines share a common coordinate pair.
Illustration: Imagine two lines on a graph. The intersection point would be the point where the two lines cross each other.
2) System of linear equations: A system of linear equations is a set of two or more equations that are solved simultaneously. The variables in these equations represent unknown values, and the goal is to find values that satisfy all of the equations in the system.
Illustration: For example, consider the system of linear equations:
x + y = 4
2x - y = 1
This system consists of two equations that are to be solved at the same time.
3) Solution of a system as an ordered pair: A solution of a system of linear equations is a set of values for the variables that makes all of the equations in the system true. In a system of two equations, the solution is often expressed as an ordered pair (x, y), where x represents the value of one variable and y represents the value of the other variable.
Illustration: Consider the system of linear equations mentioned before:
x + y = 4
2x - y = 1
The solution to this system can be represented as the ordered pair (3, 1), where x = 3 and y = 1.
1) Intersection point of two lines:
The intersection point of two lines is the point where the two lines cross or meet on a coordinate plane. It represents the common solution to the equations of the two lines. The intersection point can have various forms, such as a single point, no intersection, or overlapping lines.
For example, consider two lines: y = 2x + 1 and y = -3x + 2. By solving these equations simultaneously, we can find the intersection point(s). In this case, the intersection point is (1, 3) as both lines intersect at this coordinate.
2) System of linear equations:
A system of linear equations is a set of two or more equations with the same variables. The goal is to find values for the variables that satisfy all the equations in the system simultaneously. These equations can represent relationships or constraints in real-world problems.
For instance, consider the system of equations:
2x + 3y = 10
4x - y = 3
This is a system of two linear equations with two variables, x and y. The objective is to determine values for x and y that satisfy both equations.
3) Solution of a system as an ordered pair:
The solution of a system of linear equations can be represented as an ordered pair (x, y) that satisfies all the equations in the system. It is the common solution that makes all the equations true.
Using the previous example of the system:
2x + 3y = 10
4x - y = 3
The solution of this system is the ordered pair (2, 4), as substituting these values into both equations results in true statements:
2(2) + 3(4) = 10 (equation 1)
4(2) - 4 = 3 (equation 2)
Both equations are simultaneously satisfied by (2, 4), making it the solution of the system.
1) Intersection point of two lines: The intersection point of two lines is the point where they meet or cross each other on a graph. It is the common point between the two lines, representing the coordinates where both lines share the same values for x and y. This point is often denoted as (x, y), where x and y are the coordinates of the intersection.
To find the intersection point of two lines, you can follow these steps:
a) Express each line in the form of y = mx + c, where m is the slope of the line and c is the y-intercept.
b) Set the two equations equal to each other, so that you have an equation of the form mx + c = mx + c.
c) Solve the equation to find the value of x.
d) Substitute the value of x back into either of the original equations to find the value of y.
e) The intersection point is the coordinates (x, y) you found.
2) System of linear equations: A system of linear equations is a collection of two or more equations involving the same variables. These equations are usually linear in nature, meaning that the highest exponent in any variable is 1. The goal is to find a solution that satisfies all the equations in the system simultaneously.
To solve a system of linear equations, you have different methods available, including:
a) Substitution method: Solve one equation for one variable and substitute the expression into the other equation(s), then solve for the remaining variable(s).
b) Elimination method: Multiply one or both equations by appropriate constants to make the coefficients of one variable cancel out when the equations are added or subtracted.
c) Matrix method: Represent the system as a matrix and use matrix operations for solving the system.
3) Solution of a system as an ordered pair: The solution of a system of linear equations is the set of values that satisfy all the equations in the system at the same time. In the context of two-variable systems, the solution is often represented as an ordered pair (x, y) that corresponds to the values of the variables that make all the equations true.
For example, if you have a system of equations:
Equation 1: 3x - 2y = 5
Equation 2: 2x + y = 7
The ordered pair (2, 3) would be the solution if substituting x = 2 and y = 3 into both equations satisfies the equality of the left and right sides. Thus, when x = 2 and y = 3, Equation 1 becomes 3(2) - 2(3) = 6 - 6 = 0, and Equation 2 becomes 2(2) + 3 = 4 + 3 = 7, which are both true.