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Linear Equations

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Linear Equations

1. 1. Linear Equations <ul><li>Objectives </li></ul><ul><ul><li>To solve linear equations in one unknown. </li></ul></ul><ul><ul><li>To transpose and solve formulae. </li></ul></ul><ul><ul><li>To construct linear equations. </li></ul></ul><ul><ul><li>To solve simultaneous linear equations by substitution and elimination methods. </li></ul></ul><ul><ul><li>To use linear equations to solve problems. </li></ul></ul>
2. 2. Linear Equations <ul><ul><li>A linear equation has a variable whose value is unknown. The power of the variable is 1. </li></ul></ul><ul><ul><li>For example 7 x – 3 = 6. </li></ul></ul><ul><ul><li>4 x 2 + 5 = 12 is not linear. It is a quadratic equation because x has a power of 2. </li></ul></ul>
3. 3. Linear Equations <ul><ul><li>There are two important rules to remember when solving equations: </li></ul></ul><ul><ul><ul><li>Undo what has been done </li></ul></ul></ul><ul><ul><ul><li>Do the same to both sides of the equations </li></ul></ul></ul><ul><ul><li>Two other ‘rules’: </li></ul></ul><ul><ul><ul><li>Write your x ’s like two ‘c’s back-to back. Not like this: x </li></ul></ul></ul><ul><ul><ul><li>Write the equation out and show working, keeping the ‘=‘ in a column. </li></ul></ul></ul>
4. 7. Simultaneous Equations <ul><ul><li>A linear equation with two unknowns has an infinite number of solutions. </li></ul></ul><ul><ul><li>If we plotted them on a graph we would have a straight line. </li></ul></ul><ul><ul><li>If we plotted two of these straight line graphs on the same set of axes, they would intersect at one point provided the lines are not parallel. </li></ul></ul><ul><ul><li>Hence, there is one pair of numbers which solve both equations simultaneously. </li></ul></ul><ul><ul><li>We will look at two methods for solving </li></ul></ul><ul><li>simultaneous equations. </li></ul>
5. 8. Solving simultaneous equations by substitution <ul><ul><li>Solve the equations 2 x − y = 4 and x + 2 y = −3 using substitution </li></ul></ul><ul><ul><li>2 x − y = 4 (1) First label your equations (1) and (2) </li></ul></ul><ul><ul><li>x + 2 y = −3 (2) </li></ul></ul><ul><ul><li>Write one unknown from either equation in terms of the other unknown. </li></ul></ul><ul><ul><li>Rearranging equation (2) we get x = −3 − 2 y . </li></ul></ul><ul><ul><li>Then substitute this expression into equation (1). </li></ul></ul><ul><ul><li>2(−3 − 2 y ) − y = 4 </li></ul></ul><ul><ul><li>− 6 − 4 y − y = 4 </li></ul></ul><ul><ul><li>− 5 y = 10 </li></ul></ul><ul><ul><li>y = −2 </li></ul></ul><ul><ul><li>Substituting the value of y into (2) gives us x + 2(−2) = −3 </li></ul></ul><ul><ul><li>x = 1 </li></ul></ul><ul><ul><li>Check this answer solves both equations </li></ul></ul><ul><ul><li>This means that the point (1, –2) is the point of intersection of the graphs of the two linear relations. </li></ul></ul>
6. 9. Solving simultaneous equations by elimination <ul><ul><li>Solve the equations 2 x − y = 4 and x + 2 y = −3 using elimination </li></ul></ul><ul><ul><li>2 x − y = 4 (1) Again, label your equations (1) and (2) </li></ul></ul><ul><ul><li>x + 2 y = −3 (2) </li></ul></ul><ul><ul><li>If the coefficient of one of the unknowns in the two equations is the same, we can eliminate that unknown by subtracting one equation from the other. It may be necessary to multiply one of the equations by a constant to make the coefficients of x or y the same for the two equations. </li></ul></ul><ul><ul><li>To eliminate x , multiply equation (2) by 2 and subtract the result from equation (1). </li></ul></ul><ul><ul><li>Equation (2) becomes 2 x + 4 y = −6. Call this new equation (3) </li></ul></ul><ul><ul><li>2 x − y = 4 (1) </li></ul></ul><ul><ul><li>2 x + 4 y = −6 (3) </li></ul></ul><ul><ul><li>− 5 y = 10 (1) − (3) </li></ul></ul><ul><ul><li>y = −2 </li></ul></ul><ul><ul><li>Now substitute for y in equation (1) to find x , and check as in substitution method. </li></ul></ul>
7. 10. <ul><ul><li>For more help understanding the substitution method, try </li></ul></ul><ul><ul><li>http://www.youtube.com/watch?v=8ockWpx2KKI </li></ul></ul><ul><ul><li>For more help understanding the elimination method, try </li></ul></ul><ul><ul><li>http://www.youtube.com/watch?v=XM7Q4Oj5OTc </li></ul></ul>
8. 13. Rearranging Formulae