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The solutions to many problems may be found by translating the problems into mathematical equations which may then be solved using algebraic techniques. In this case there is no other value of x that would give a true statement. In this chapter methods of solving linear equations and linear simultaneous equations using a C omputer A lgebra S ystem will be included. It is most important that the steps taken to solve the equation are done in the correct order.
Subtract 4 from both sides. Divide both sides by 3. Once a solution has been found it may be checked by substituting the value back into both sides of the original equation to ensure that the left-hand side LHS equals the right-hand side RHS. Group all the terms containing the unknown on one side of the equation and the remaining terms on the other. Chapter 1 — Reviewing Linear Equations. First remove the brackets and then use the previous rules. Solve x. Multiply both sides of the equation by the lowest common multiple of 3 and 5.
Remember that the line separating the numerator and the denominator the vinculum acts as a bracket. Multiply by 6, the lowest common denominator. Using a CAS calculator. From the Home screen press Algebra menu. The 1:solve menu is used to give the. Literal equations can be solved with a CAS calculator. Be careful to put in a multiplication sign between the letters of the expression as otherwise it will read them as a single variable rather than a product. From the Home screen press.
Solve each of the following equations for x :. By representing the unknown quantity in a problem with a symbol and constructing an equation from the information, the value of the unknown can be found by solving the equation. Before constructing the equation, each symbol and what it stands for including the units should be stated. It is essential to remember that all the elements of the equation must be in units of the same system.
If the chef forgot to weigh a turkey before cooking it, but knew that it had taken 3 hours to. The turkey weighed 5. Adam normally takes 5 hours to travel between Higett and Logett. Find his normal speed. For each of the following write an equation using the pronumeral x then solve the equation for x :. If B receives three times as much as A and C receives twice as much as A , how much does each receive? The sum of two numbers is 42, and one number is twice the other.
Find the two numbers. Find the area of a rectangle whose perimeter is 4. Find three consecutive whole numbers with a sum of Find four consecutive odd numbers with a sum of Two tanks contain equal amounts of water. They are connected by a pipe and litres of water is pumped from one tank to the other. One tank then contains 6 times as much water as the other. How many litres of water did each tank contain originally?
A page book has p lines to a page. If the number of lines were reduced by three on each page the number of pages would need to be increased by 20 to give the same amount of writing space. How many lines were there on the page originally? A rower travels upstream at 6 km per hour and back to the starting place at 10 km per hour.
The total journey takes 48 minutes. How far upstream did the rower go? How many dozens were there in each of the crates?
At the end of that time she was 6 km from the starting point. Find the value of x. The total journey takes 45 minutes. Find the distance travelled. In 2 years time the sum of their ages will be Find the present ages of father and son. A linear equation that contains two unknowns, e. Such an equation actually expresses a relationship between pairs of numbers, x and y , that satisfy the equation. If all the possible pairs of numbers x , y that will satisfy the equation.
Hence the name linear relation. If the graphs of two such equations are drawn on the same set of axes, and they are. Hence there is one pair of numbers that will satisfy both equations simultaneously. Finding the intersection of two straight lines can be done graphically; however the accuracy. Alternatively this point of intersection may be found algebraically by solving the pair of. Three techniques for solving simultaneous equations will be considered. First express one unknown from either equation in terms of the other unknown.
Then substitute this expression into the other equation. Equation 1 then becomes Solving 1. Substituting the value of y into 2.
Check in 1 :. Note: This means that the point 1, —2 is the point of intersection of the graphs of the two linear relations. To eliminate x , multiply equation 2 by 2 and subtract the result from equation 1.
Equation 2 becomes. The and can be typed or. The simultaneous equations can also be solved. The equations are rearranged to make. The equations in this form are. Graph them by pressing. Follow the prompts.
For 1st curve? For 2nd curve? For Lower Bound? For Upper Bound? The result is shown. Solve each of the following pairs of simultaneous equations by the substitution method:. Let x and y be the two numbers. Add equations 1 and 2. This gives. Substitute in equation 1. Check in 2 :. Find the cost of 1 kilogram of jam and 1 kilogram of butter. Multiply 1 by Subtract 1 from 2 :. Substituting in 2 :.
Check in the original problem:. Find two numbers whose sum is and whose difference is How much does one stool cost? How much does one belt cost?
Use simultaneous equations to solve the following. Find a pair of numbers whose sum is 45 and whose difference is In four years time a mother will be three times as old as her son. Find their present ages. A party was organised for thirty people at which they could have either a hamburger or a pizza.
Essential Mathematical Method 1 & 2 CAS Chapter 1