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Connecting Algebra with the Coordinate Plane

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The year is Lying on his back, French mathematician René Descartes, watches a fly crawl across the ceiling. Suddenly, an idea comes to him. He visualizes two number lines, intersecting at a 90° angle. He realizes that he can graph the fly's location on a piece of paper. Descartes called the main horizontal line the x-axis and the main vertical line the y-axis. He named the point where they intersect the origin.

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**Descartes represented the fly's location as an ordered pair of numbers.**

The first number, the x-value, is the horizontal distance along the x-axis, measured from the origin. The second number, the y-value, is the vertical distance along the y-axis, also measured from the origin.

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Y-COORDINATE The locations in the plane where the x and y values intersect are called coordinates. ORIGIN X-COORDINATE In the image on the right, three points — (0, 0), (-2, 5), and (3, -6.5) — are represented as graphed coordinates.

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The plane containing these points is called the Cartesian plane (in honor of Descartes), or the coordinate plane. Together, the x-axis, the y-axis, the coordinate plane, and all the coordinates make up the Cartesian coordinate system. CARTESIAN COORDINATE SYSTEM

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The plane determined by a horizontal number line, called the x-axis, and a vertical number line, called the y-axis, intersecting at a point called the origin. Each point in the coordinate plane can be specified by an ordered pair of numbers. COORDINATE PLANE The point (0, 0) on a coordinate plane, where the x-axis and the y-axis intersect. ORIGIN

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Y-COORDINATE ORIGIN X-COORDINATE CARTESIAN COORDINATE SYSTEM

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**Points in Quadrant 2 have negative x but positive y coordinates. **

Points in Quadrant 1 have positive x and positive y coordinates. To make it easy to talk about where on the coordinate plane a point is, we divide the coordinate plane into four sections called quadrants. Points in Quadrant 3 have negative x and negative y coordinates. Points in Quadrant 4 have positive x but negative y coordinates.

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**Write the coordinates of each point.**

B: (5, -3) C: (-1, -5)

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**WRITE THE COORDINATES FOR EACH POINT**

B: (4, 6) C: (2, 1) D: (-4, 3) E: (-5, 0) F: (-3, -2) G: (0, -4) H: (4, -4)

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**EVERY POINT THAT FITS ON A LINE IS CALLED “COLLINEAR”. **

Every straight line can be represented by an equation: y = mx + b. The coordinates of every point on the line will solve the equation if you substitute them in the equation for x and y. EVERY POINT THAT FITS ON A LINE IS CALLED “COLLINEAR”. THESE POINTS ARE “COLLINEAR”. y = 2x + 1 0 = 2(1) + 1== YES 3 = 2(1) + 1== YES 5 = 2(2) + 1== YES

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**All segments shown are parallel to either the x- or y-axis**

All segments shown are parallel to either the x- or y-axis. Determine the ordered pair that represents each point. J: ( , ) – 6 12 G: ( , ) – 6 B: ( , ) 1 N: ( , ) 12 F: ( , ) 8 1 E: ( , ) 8 – 2

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**Determine whether the following points are “collinear” to “G” and “H”.**

POINTS G(1, – 1) AND H(3, 3) LIE ON THE LINE “y = 2x – 3”. Determine whether the following points are “collinear” to “G” and “H”. I (0, – 3) y = 2x – 3 – 3 = 2(0) – 3 YES J (2, 1) 1 = 2(2) – 3 YES K (– 3, –8) –8 = 2(–3) – 3 NO L ( 5, 7) 7 = 2(5) – 3 YES 16 = 2(10) – 3 NO M ( 10, 16)

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