High School Modules > Algebra by Gregory A. Moore
Solving Linear Equations
Learning the steps to solve linear equations.
[Directions : Execute the Code Resource section first. Although there will be no output immediately, these definitions are used later in this worksheet.]
Code
| > | restart; |
| > | AddBothSides := proc( expr, a ) lhs(expr) + a = rhs( expr) + a; end proc: |
| > | MultBothSides := proc( expr, a ) simplify(lhs(expr) * a) = simplify( rhs( expr) * a) ; end proc: |
| > | SimplifyBothSides := proc( expr ) simplify( factor( lhs(expr) )) = simplify( factor( rhs(expr) )); end proc: |
Example 1
| > | 17*x - 5 = 29; |
Lets get rid of the number term, -5, on the same side as the x term, by adding its opposite, +5
to both sides.
| > | AddBothSides( %, 5); |
Now, we multiply both sides by the reciprocal of 17.
| > | MultBothSides( %, 1/17); |
We are done. We found the answer. Maple can also solve the problem in one step.
| > | solve( 17*x - 5 = 29, x) ; |
Example 2
| > | 1/7*x+1 = 1/4*x-35; |
Lets first clear the denominators by multiplying both sides by the LCM, 28, of the
denominators, 7 and 4.
| > | MultBothSides( %, 28); |
Now lets get rid of the smaller x term, so only one x term remains.
| > | AddBothSides( %, -4*x); |
We still have two purely number terms, 28 and 980. Lets eliminate the one, -980, on the same side as the
x term, by adding its opposite to both sides.
| > | AddBothSides( %, 980); |
There is one final step : to change 3x into x by multiplying y by the reciprocal, 1/3.
| > | MultBothSides( %, 1/3); |
We are done. We found he answer. Maple can also solve the problem in one step.
| > | solve( 1/7*x+1 = 1/4*x-35, x) ; |
Note that we added to both sides twice in a row. First we added -4x, then we added 980. Why not
do both steps at once? Here is how this problem would look if we sped it up in this way.
| > | 1/7*x+1 = 1/4*x-35; |
| > | MultBothSides( %, 28); |
| > | AddBothSides( %, -4*x + 980 ); |
| > | MultBothSides( %, 1/3); |
Example 3
| > | (3/5)*(x - 7/3) = (1/9)*x + 3*(4*x - 5/2); |
Note that Maple automatically distributes and combines like terms on both sides.
However, there is still more to do. Lets first clear the denominators by multiplying
both sides by the LCM, 90, of the various denominators, 2, 5 and 9.
| > | MultBothSides( %, 90); |
Now lets get rid of the smaller x term, so only one x term remains.
| > | AddBothSides( %, -54*x); |
We still have two purely number terms. Lets eliminate the one, -675, on the same side as the
x term, by adding its opposite, +675, to both sides.
| > | AddBothSides( %, 675); |
There is one final step : to change 1036x into x by multiplying y by the reciprocal, 1/1036.
| > | MultBothSides( %, 1/1036); |
We are done. We found the answer. Maple can also solve the problem in one step.
| > | solve( (3/5)*(x - 7/3) = (1/9)*x + 3*(4*x - 5/2), x) ; |
Example 4
These same methods will be used to solve more advanced problems, later. For example, if we have
an expression like this :
| > | 3/(x-2) + 5/x = 7; |
We can multiply both sides by the LCM of the algebraic expressions in the denominators.
| > | MultBothSides( %, (x-2)*x ); |
In this case, we can move everything to one side by adding the opposite of (8x-10) to both sides.
| > | AddBothSides( %, -(8*x -10) ); |
We can simplify both sides.
| > | SimplifyBothSides(%); |
Since this is not a linear equation, we need to use other methods. We'll discuss this more later.
| > | solve(%); |
| > |
| > |
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