Solving linear equations: Section 2
If equal terms appear on both sides of an equation,
x + b + d = c + d.
d appears on both sides. Therefore, we may cancel them.
x + b = c.
Theoretically, we can say that we subtracted d from both sides.
Finally, on solving for x:
x = c − b.
Problem 17. Solve for x :
x² + x − 5 = x² − 3
To see the answer, pass your mouse from left to right
Cancel the x²'s:
Problem 18. Solve for x :
x − a + b = a + b + c
Cancel the b's but not the a's. On the left is −a, but on the right is +a. They are not equal.
The unknown on both sides
Example 3. Solve for x :
This is another example of doing algebra with your eyes. In the first line, you should see that 2x goes to the left as −2x, and that −3 goes to the right as +3.
As a general rule for solving any linear equation, we can now state the following:
Transpose all the terms that involve the unknown to the left, and add them;
Problem 19. Solve for x :
Problem 20. Solve for x :
Problem 21. Remove parentheses, add like terms, and solve for x :
Simple fractional equations
Since 2 divides on the left, it will multiply on the right:
Problem 22. Solve for x :
Example 5. Solve for x:
Solution. In the standard form of a simple fractional equation, x is in the numerator. But we can easily make that standard form by taking the reciprocal of both sides.
For if two numbers are equal, then their reciprocals are equal.
Example 6. Fractional coefficient.
Since 4 divides on the left, it will multiply on the right:
And since 3 multiplies on the left, it will divide on the right:
In other words, goes to the other side as its reciprocal, .
Coefficients go to the other side as their reciprocals.
The reciprocal of ½ is 2.
Problem 29. The Celsius temperature C is related to the Fahrenheit temperature F by this formula,
b) Solve the formula for C.
This is Problem 13 of Lesson 1.
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