Chapter 1 Linear Equations PDF

Linear Equations CHAPTER 1: LINEAR EQUATIONS A linear equation is an equation for a straight line. Linear equations can have one or more variables. Maximum power of a variable in a linear equation is 1. Examples of linear equations: 𝑦 = 5𝑥 − 2 𝑦 − 4 = 6(𝑥 + 2) 𝑦 + 3𝑥 − 7 = 0 3𝑥 =...

Linear Equations CHAPTER 1: LINEAR EQUATIONS A linear equation is an equation for a straight line. Linear equations can have one or more variables. Maximum power of a variable in a linear equation is 1. Examples of linear equations: 𝑦 = 5𝑥 − 2 𝑦 − 4 = 6(𝑥 + 2) 𝑦 + 3𝑥 − 7 = 0 3𝑥 = 8 𝑦 5 = 9 Examples of NOT linear equations: 𝑦2 + 5 = 0 2√𝑥 − 𝑦 = 8 𝑥3 3 = 12 1.1 Rectangular Coordinates The horizontal axis is usually called the x axis and the vertical axis is usually called the y axis. The point where the axes cross is usually chosen to be where x = 0 and where y = 0; this point is called the origin. The axes divide the plane into four regions called quadrants, which are numbered counterclockwise. For example the point P shown to the right has an x coordinate of 3 and a y coordinate of 5. We say that this point is at x = 3 and y = 5 and we use the ordered pair notation (3, 5) to describe this point. 1  Linear Equations 1.2 Slope of a line, m The slope (also called gradient) of a straight line shows how steep a straight line is. Slope is an important characteristic of a line. This can be easily seen using rectangular coordinates. Let P = ( x1 , y1 ) and Q = (x2 , y 2 ) be two different points where (x1 , y1 )  (x2 , y 2 ). The slope, denoted as m of a line that contains point P and Q is obtained using the formula y 2 − y1 m= x2 − x1 Q P m = undefined m=0 1.3 Intercept of a line The x-intercept of a line is the point at which the line crosses the x axis, where the y value equals to zero. The x-intercept = (x,0 ). The y-intercept of a line is the point at which the line crosses the y axis, where the x value equals to zero. The y-intercept = (0, y ). Example 1.1: Find the slope, y-intercept and the x-intercept of the line 2 x + 4 y − 8 = 0. 2  Linear Equations 1.4 Forms of linear equations Three different forms of linear equations: (i) General form (ii) Point-slope form (iii) Slope-intercept form 1.4.1. General Form A linear equation in two variables can be written in the form Ax + By = C where A, B, C → constants x, y → variables A, B  0. Examples 1.2: Find the values of A, B and C. i. 3x − 5 y − 6 = 0 This equation can be written as 3x − 5 y = 6.  A = 3, B = −5 & C = 6. ii. − 3x = 2 y − 1 This equation can be written as − 3x − 2 y = −1.  A = −3, B = −2 & C = −1. iii. x=4 This equation can be written as x + 0 y = 4.  A = 1, B = 0 & C = 4. 3  Linear Equations 1.4.2. Point-slope form Given that point P = ( x1 , y1 ) lies on a line with slope m and the equation of the straight line in the point slope form can be written as y − y1 = m(x − x1 ) where 𝑃(𝑥1 , 𝑦1 ) = point lies on the line 𝑥, 𝑦 = variables 𝑚 = slope Example 1.3: Given 2 points 𝐴(1,2) and 𝐵(3,4). Find the equation of 𝐴𝐵 using the point slope formula. 1.4.3. Slope-intercept form Given slope m, y-intercept (0, c ) and any other point on the line, say Q( x, y ). Using the point slope form, we have y = mx + c where 𝑥, 𝑦 = variables 𝑚 = slope 𝑐 = 𝑦 − intercept Example 1.4: Given 2 points 𝐴(−1,2) and 𝐵(3, −5). Find the equation of 𝐴𝐵 using the slope intercept formula. 4  Linear Equations 1.5 Graph When graphing, it is important to find the intercepts. Find the x-intercept (the value of x when y = 0) and the y intercept (the value of y when x = 0). Plot the intercepts on the xy plane. Draw a line to connect the points. Example 1.5: Graph the straight line 2 x + 3 y = 6. Example 1.6: Sketch the following equations. i. y = −2 ii. x=3 iii. 2x + y = 4 iv. 2x − y = 8 5  Linear Equations 1.6 Solving linear equations 1.6.1 Method of Substitution Steps for Solving by Substitution 1. Pick one of the equations and solve for one of the variables in terms of the remaining variables. 2. Substitute the result in the remaining equations. 3. If one equation in one variable results, solve this equation. Otherwise, repeat step 1 until a single equation with one variables remains. 4. Find the values of remaining variables by back-substitution. 5. Check the solution found. Example 1.7: Use method of substitution to solve the system of equation 𝑥 + 𝑦 = −1 { 4𝑥 − 3𝑦 = 10 6  Linear Equations 1.6.2 Method of Elimination This method is usually preferred over substitution if substitution leads to fractions or if the system contains more than two variables. The idea behind the method of elimination is to keep replacing the original equations in the system with equivalent equations until a system of equations with an obvious solution is reached. Rules for Obtaining an Equivalent System of Equations: 1. Interchange any two equations of the system. 2. Multiply (or divide) each side of an equation by the same nonzero constant. 3. Replace any equation in the system by the sum (or difference) of that equation and any other equation in the system. Example 1.8: Use the method of elimination by addition to solve the system of equations. 𝑥 + 𝑦 = −1 { 4𝑥 − 3𝑦 = 10 7  Linear Equations 1.7 Pair of linear equations (i) Coincident – All points on L are the same as the points on M. These lines are identical. (ii) Parallel – L and M have no points in common at all and they have the same slope. m1=m2 (iii) Intersecting – L and M have exactly one point in common. The intersection point can be found by solving simultaneous equations. 8  Linear Equations Example 1.9: Show that L : 2 x + 3 y = 6 and M : 4 x + 6 y = 0 are parallel. y=mx+c m1=m2 L:3y=-2x+6 y=-2/3x+2 M:6y=-4x+0 y=-4/6x+0/6 y=-2/3x+0 Example 1.10: Find the intersection point of L : 2 x − y = 5 and M : 4 x + 6 y = 0. L: 2x-5=y 4x+6(2x-5)=0 4x+12x-30=0 16x=30 x=15/8 (15/8,-5/4) Example 1.11: Given that L : x − 4 y = 8. Find equation of the line that passes through (2,1) and parallel to L. 9  Linear Equations Tutorial 1 1. Determined the slope of the line that links the two coordinates a) (2,5) and (-2,13) c) (-1,-7) and (3,3) b) (3,1) and (3,5) d) (-8,4) and (8,4) 2. Find the slope and y-intercept of each line a) 3𝑥 + 4𝑦 = 20 b) 𝑥 – 2𝑦 = 10 3. Write the equation of each line in the form general form. 1 a) Slope = 2; (−2, 1) c) Slope = − 2 ; (6, −3) b) (2, −4) and (5, 7) d) (1,8) and (3,8) 4. Find the axis intercepts of the line a) 3𝑥 + 2𝑦 = 12 b) 4𝑥 − 3𝑦 = 24 5. What is the equation of the x-axis? Sketch 2 other lines that are parallel to the x-axis. State the equation of each line. What are their slopes? horizontal 6. Sketch the line passing through point (5, 0) with slope equals to zero and state its equation. 7. Given that the point P(2,5) lies on the line kx + 3 y + 9 = 0. Find k. 8. Determine whether the line AB is parallel to the line CD. (a) A(1,-2), B(-3,-10) and C(1,5), D(-1,1) (b) A(2,3), B(2,-2) and C(-2,4), D(-2,5) 9. Find equation of each line : m 2 (a) Passes through point P(-1,4) and has slope of. 3 (5,0) (b) Passes through point P(-1,-6) and has an x intercept 5. 10  Linear Equations 10. Find the intersection point of x + y = 5 and 3x − y = 7. 11. Find the equation of the (a) vertical line that passes through (0,5). (b) horizontal line that passes through (-4,-3). (c) line that passes through the point (0, 0) and has the slope zero. (d) line passes through the points (2,1) and (2,5). (e) line if the line is parallel to x-axis and 10 units below it. 12. Find the equation for the line with properties below: (a) Parallel to the line y = 3x passing through (-1,2). (b) Parallel to the line 2 x − y + 2 = 0 passing through (0,0). (c) Parallel to the line x = 5 passing through (4,2). x y 13. Find the equation of the line passing through (-5,-4) and parallel to the line passing through (-3,2), and (6,8). 14. Find the value of a so that the line passing through the points (a,2) and (3,6) is parallel to a line with slope 4. m=4 15. If the line passing through the points (1,a) and (4,-2) is parallel through the points (2,8) and (-7,a+4), what is the value of a? 16. Find A so that Ax − 4 y + 3 = 0 is parallel to the line 2 x + 2 y − 5 = 0. 17. A college student receives an interest free loan of $8250 from a relative. The student will pay $125 a month until the loan is paid off. time=number of months loan (a) Express the amount, P remaining to be paid in terms of t. (b) After how many months, the student will owe $5000. (c) Sketch a graph showing the relationship between P and t for the duration of the loan. 18. A rock concert brought in RM432500 on the sale of 9500 tickets. If the tickets sold for RM35 and RM55 each, how many of each type of ticket were sold? 11  Linear Equations Answers 1. (a) -2 (b) ∞ (c) 5⁄2 (d) 0 3 1 2. (a) 𝑦 = − 4 𝑥 + 5 (b) 𝑦 = 2 𝑥 − 5 3. (a) −2𝑥 + 𝑦 = 5 (b) −11𝑥 + 3𝑦 = −34 (c) 𝑥 + 2𝑦 = 0 (d) 𝑦 = 8 4. (a) x-int = 4 and y-int = 6 (b) x-int = 6 and y-int = -8 5. y = 0, y = 1, y = -2, slope, m = 0. 6. y=0 7. k = -12 8. (a) parallel (b) parallel 9. (a )y = 2 x + 14 (b) y = x−5 3 3 10. (3,2) 11. (a) x = 0 (b) y = -3 (c) y = 0 (d) x = 2 (e) y = -10 12. (a) y = 3x + 5 (b) y = 2 x (c) x = 4 2 2 13. y = x − 3 3 14. a = 2 15. a = -5 16. A = - 4 17. (a) P = 8250− 125t (b) 26 months 18. 4500 RM35 tickets and 5000 RM55 tickets. 12