Topic 4.2.1

1. Points on a Line Topic 4.2.1

2. Topic 4.2.1 Points on a Line California Standards: 6.0 Students graph a linear equationand compute the x- and y-intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the region defined by 2x + 6y < 4). 7.0 Students verify that a point lies on a line, given an equation of the line.Students are able to derive linear equations by using the point-slope formula. What it means for you: You’ll learn how to show mathematically that points lie on a line. • Key Words: • linear equation • variable • solution set • verify

3. (3, 5) (2, 3) (1, 1) (–1, –3) Topic 4.2.1 Points on a Line You already dealt with lines in Topics 4.1.3 and 4.1.4. In this Topic you’ll see a formal definition relating ordered pairs to a line — and you’ll also learn how to show that points lie on a particular line.

4. Topic 4.2.1 Points on a Line Graphs of Linear Equations are Straight Lines An equation is linearif the variables have an exponent of one and there are no variables multiplied together. For example: Linear: 3x + y = 4, 2x = 6, y = 5 – x Nonlinear:xy= 12, x2 + 3y = 1, 8y3 = 20

5. Topic 4.2.1 Points on a Line Linear equations in two variables, x and y, can be written in the form: Ax + By = C The solution setto the equation Ax + By = C consists of all ordered pairs (x, y) that satisfy the equation. All the points in this solution set lie on a straight line. This straight line is the graphof the equation.

6. If the ordered pair (x, y) satisfies the equation Ax + By = C, then the point (x, y) lies on the graph of the equation. The ordered pair (–2, 5) satisfies the equation 2x + 2y = 6. So the point (–2, 5) lies on the line 2x + 2y = 6. The point (2, 1) lies on the line 2x + 2y = 6. So the ordered pair (2, 1) satisfies the equation 2x + 2y = 6. Topic 4.2.1 Points on a Line

7. Topic 4.2.1 Points on a Line Verifying That Points Lie on a Line To determine whether a point (x, y) lies on the lineof a given equation, you need to find out whether the ordered pair (x, y) satisfies the equation. If it does, the point is on the line. You do this by substitutingxand yinto the equation.

8. Topic 4.2.1 Points on a Line Example 1 a) Show that the point (2, –3) lies on the graph of x – 3y = 11. b) Determine whether the point (–1, 1) lies on the graph of 2x + 3y = 4. Solution a) 2 – 3(–3) = 11 Substitute 2 for x and –3 for y 2 + 9 = 11 11 = 11 A true statement So the point (2, –3) lies on the graph of x – 3y = 11, since (2, –3) satisfies the equation x – 3y = 11. Solution continues… Solution follows…

9. Topic 4.2.1 Points on a Line Example 1 a) Show that the point (2, –3) lies on the graph of x – 3y = 11. b) Determine whether the point (–1, 1) lies on the graph of 2x + 3y = 4. Solution (continued) b) If (–1, 1) lies on the graph of 2x + 3y = 4, then 2(–1) + 3(1) = 4. But 2(–1) + 3(1) = –2 + 3 = 1 Since 1 ¹ 4, (–1, 1) does not lie on the graph of 2x + 3y = 4.

10. Topic 4.2.1 Points on a Line Guided Practice Determine whether or not each point lies on the line of the given equation. • (–1, 2); 2x – y = –4 • (3, –4); –2x – 3y = 6 • (–3, –1); –5x + 3y = 11 • (–7, –3); 2y – 3x = 15 yes: 2(–1) – 2 = –4 yes: –2(3) – 3(–4) = 6 no: –5(–3) + 3(–1) = –18 ¹ 11 yes: 2(–3) – 3(–7) = 15 Solution follows…

11. Topic 4.2.1 Points on a Line Guided Practice Determine whether or not each point lies on the line of the given equation. • (–2, –2); y = 3x + 4 • (–5, –3); –y + 2x = –7 • (–2, –1); 8x – 15y = 3 yes: –2 = 3(–2) + 4 yes: –(–3) + 2(–5) = –7 no: 8(–2) – 15(–1) = –1 ¹ 3 Solution follows…

12. 1 1 2 2 2 1 1 2 no: 6( ) – 16(– ) = 6 ¹ 7 4 5 3 3 3 4 3 5 yes: –3( ) – 10(– ) = 2 Topic 4.2.1 Points on a Line Guided Practice Determine whether or not each point lies on the line of the given equation. • (1, 4); 4y– 12x = 3 • ( , – ); 6x – 16y = 7 • ( , – ); –3x – 10y = 2 no: 4(4) – 12(1) = 4 ¹ 3 Solution follows…

13. Topic 4.2.1 Points on a Line Independent Practice In Exercises 1–4, determine whether or not each point lies on the graph of 5x – 4y = 20. • (0, 4) • (4, 0) • (2, –3) • (8, 5) no: 5(0) – 4(4) = –16 ¹ 20 yes: 5(4) – 4(0) = 20 no: 5(2) – 4(–3) = 22 ¹ 20 yes: 5(8) – 4(5) = 20 Solution follows…

14. Topic 4.2.1 Points on a Line Independent Practice In Exercises 5–8, determine whether or not each point lies on the graph of 6x + 3y = 15. • (2, 1) • (0, 5) • (–1, 6) • (3, –1) yes: 6(2) + 3(1) = 15 yes: 6(0) + 3(5) = 15 no: 6(–1) + 3(6) = 12¹ 15 yes: 6(3) + 3(–1) = 15 Solution follows…

15. Topic 4.2.1 Points on a Line Independent Practice In Exercises 9–12, determine whether or not each point lies on the graph of 6x – 6y = 24. • (4, 0) • (1, –3) • (100, 96) • (–400, –404) yes: 6(4) – 6(0) = 24 yes: 6(1) – 6(–3) = 24 yes: 6(100) – 6(96) = 24 yes: 6(–400) – 6(–404) = 24 Solution follows…

16. Topic 4.2.1 Points on a Line Independent Practice 13. Explain in words why (2, 31) is a point on the line x = 2, but not a point on the line y = 2. x=2includesallpointswhosex-coordinateis2,so(2,31)isapoint onx=2;they-coordinateis31,so(2,31)doesnotlieontheliney=2. 14. Determine whether the point (3, 4) lies on the lines 4x + 6y = 36 and 8x – 7y = 30. 4x + 6y = 36: 4(3) + 6(4) = 36 8x – 7y = 30: 8(3) – 7(4) = –4 ¹36 (3, 4) does lie on the line 4x + 6y = 36, but not on the line 8x – 7y = 30, so it doesn’t lie on both lines. Solution follows…

17. Topic 4.2.1 Points on a Line Round Up You can always substitute x and y into the equation to prove whether a coordinate pair lies on a line. That’s because if the coordinate pair lies on the line then it’s actually a solution of the equation.