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These techniques involve rewriting problems in the form of symbols. For example, the stated problem "Find a number which, when added to 3, yields 7" may be written as: We call such shorthand versions of stated problems equations, or symbolic sentences.
The terms to the left of an equals sign make up the left-hand member of the equation; those to the right make up the right-hand member.
The value of the variable for which the equation is true 4 in this example is called the solution of the equation.
We can determine whether or not a given number is a solution of a given equation by substituting the number in place of the variable and determining the truth or falsity of the result.
The first-degree equations that we consider in this chapter have at most one solution. The solutions to many such equations can be determined by inspection. Example 2 Find the solution of each equation by inspection. However, the solutions of most equations are not immediately evident by inspection.
Hence, we need some mathematical "tools" for solving equations. In solving any equation, we transform a given equation whose solution may not be obvious to an equivalent equation whose solution is easily noted. The following property, sometimes called the addition-subtraction property, is one way that we can generate equivalent equations.
If the same quantity is added to or subtracted from both members of an equation, the resulting equation is equivalent to the original equation.
The next example shows how we can generate equivalent equations by first simplifying one or both members of an equation. We want to obtain an equivalent equation in which all terms containing x are in one member and all terms not containing x are in the other.
Sometimes one method is better than another, and in some cases, the symmetric property of equality is also helpful. Also, note that if we divide each member of the equation by 3, we obtain the equations whose solution is also 4. In general, we have the following property, which is sometimes called the division property.
If both members of an equation are divided by the same nonzero quantity, the resulting equation is equivalent to the original equation. Solution Dividing both members by -4 yields In solving equations, we use the above property to produce equivalent equations in which the variable has a coefficient of 1.
Also, note that if we multiply each member of the equation by 4, we obtain the equations whose solution is also In general, we have the following property, which is sometimes called the multiplication property.
If both members of an equation are multiplied by the same nonzero quantity, the resulting equation Is equivalent to the original equation. Example 1 Write an equivalent equation to by multiplying each member by 6.
Solution Multiplying each member by 6 yields In solving equations, we use the above property to produce equivalent equations that are free of fractions. There is no specific order in which the properties should be applied. Any one or more of the following steps listed on page may be appropriate.
Steps to solve first-degree equations: Combine like terms in each member of an equation. Using the addition or subtraction property, write the equation with all terms containing the unknown in one member and all terms not containing the unknown in the other.
Combine like terms in each member. Use the multiplication property to remove fractions. Use the division property to obtain a coefficient of 1 for the variable. We can solve for any one of the variables in a formula if the values of the other variables are known. We substitute the known values in the formula and solve for the unknown variable by the methods we used in the preceding sections.
Solution We can solve for t by substituting 24 for d and 3 for r. We use the same methods demonstrated in the preceding sections. Solution We may solve for t in terms of r and d by dividing both members by r to yield from which, by the symmetric law, In the above example, we solved for t by applying the division property to generate an equivalent equation.
Sometimes, it is necessary to apply more than one such property. Solution We can solve for x by first adding -b to each member to get then dividing each member by a, we have.Math Playground has hundreds of interactive math word problems for kids in grades Solve problems with Thinking Blocks, Jake and Astro, IQ and more.
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