Key Concepts
- Ordered Pair An ordered pair, [latex]\left(x,y\right)[/latex] gives the coordinates of a point in a rectangular coordinate system.[latex]\begin{array}{c}\text{The first number is the }x\text{-coordinate}.\hfill \\ \text{The second number is the }y\text{-coordinate}.\hfill \end{array}[/latex]
- Steps for Plotting an Ordered Pair (x, y) in the Coordinate Plane
- Determine the x-coordinate. Beginning at the origin, move horizontally, the direction of the x-axis, the distance given by the x-coordinate. If the x-coordinate is positive, move to the right; if the x-coordinate is negative, move to the left.
- Determine the y-coordinate. Beginning at the x-coordinate, move vertically, the direction of the y-axis, the distance given by the y-coordinate. If the y-coordinate is positive, move up; if the y-coordinate is negative, move down.
- Draw a point at the ending location. Label the point with the ordered pair.
- An ordered pair is represented by a single point on the graph.
- Sign Patterns of the Quadrants
Quadrant I Quadrant II Quadrant III Quadrant IV [latex](x,y)[/latex] [latex](x,y)[/latex] [latex](x,y)[/latex] [latex](x,y)[/latex] [latex](+,+)[/latex] [latex](−,+)[/latex] [latex](−,−)[/latex] [latex](+,−)[/latex] - Coordinates of Zero
- Points with a [latex]y[/latex]-coordinate equal to [latex]0[/latex] are on the x-axis, and have coordinates [latex] (a, 0)[/latex].
- Points with a [latex]x[/latex]-coordinate equal to [latex]0[/latex] are on the y-axis, and have coordinates [latex](0, b)[/latex].
- The point [latex](0, 0)[/latex] is called the origin. It is the point where the x-axis and y-axis intersect.
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Identifying Solutions To find out whether an ordered pair is a solution of a linear equation, you can do the following:
- Graph the linear equation, and graph the ordered pair. If the ordered pair appears to be on the graph of a line, then it is a possible solution of the linear equation. If the ordered pair does not lie on the graph of a line, then it is not a solution.
- Substitute the (x, y) values into the equation. If the equation yields a true statement, then the ordered pair is a solution of the linear equation. If the ordered pair does not yield a true statement then it is not a solution.
Glossary
- linear equation
- An equation of the form [latex]Ax+By=C[/latex], where [latex]A[/latex] and [latex]B[/latex] are not both zero, is called a linear equation in two variables.
- ordered pair
- An ordered pair [latex]\left(x,y\right)[/latex] gives the coordinates of a point in a rectangular coordinate system. The first number is the [latex]x[/latex] -coordinate. The second number is the [latex]y[/latex] -coordinate.
[latex]\underset{x\text{-coordinate},y\text{-coordinate}}{\left(x,y\right)}[/latex]
- origin
- The point [latex]\left(0,0\right)[/latex] is called the origin. It is the point where the the point where the [latex]x[/latex] -axis and [latex]y[/latex] -axis intersect.
- quadrants
- The [latex]x[/latex] -axis and [latex]y[/latex] -axis divide a rectangular coordinate system into four areas, called quadrants. The quadrants are labeled with the Roman Numerals I, II, III, IV going around the coordinate system in a counter-clockwise direction.
- solution to a linear equation in two variables
- An ordered pair [latex]\left(x,y\right)[/latex] is a solution to the linear equation [latex]Ax+By=C[/latex], if the equation is a true statement when the x- and y-values of the ordered pair are substituted into the equation.
- x-axis
- The x-axis is the horizontal axis in a rectangular coordinate system.
- y-axis
- The y-axis is the vertical axis on a rectangular coordinate system.
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