Problem 109 - Introductory Algebra for College Students

Question

determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The ordered pair \((3,4)\) satisfies the equation $$ 2 y-3 x=-6 $$

Step-by-Step Solution

Answer

After substituting and simplifying, the statement 'The ordered pair (3,4) satisfies the equation \(2y - 3x = -6\)' is false. An accurate version of the statement would be 'The ordered pair (3,0) satisfies the equation \(2y - 3x = -6\)'.

Step 1: Substitution of the values

Begin by substituting the x value (which is 3) and the y value (which is 4) into the given equation, that provides \(2*4 - 3*3 = -6\).

Step 2: Solving the Equation

After carrying out the operations, the left side of the equation becomes \(8 - 9 = -1\).

Step 3: Result Verification

Since -1 does not equal -6, the original statement is false. Here, the verification process is important to confirm the accuracy (or in this case, the inaccuracy) of the original statement.

Step 4: Adjustment

For the equation to be correct, the pair can be altered. A true statement would be (3,0) satisfies the equation \(2y - 3x = -6\). With y=0 and x=3, this plugging in results in \(-9 = -9\), which is obviously true.

Key Concepts

Ordered Pairs

An ordered pair consists of two elements written in a specific sequence, usually as \(x, y\). It is a way to identify positions on a graph or within an equation.
For example, the ordered pair \(3, 4\) assigns 3 to \(x\) and 4 to \(y\).

• Ordered pairs are essential in graphing and solving equations since they establish exact relationships between variables.
• When substituting into an equation, the order is vital; the first number corresponds to the \(x\) value, and the second to the \(y\) value.

This structure helps you determine whether certain conditions or solutions apply to particular functions or graphs.

True or False Statements

Determining whether a statement is true or false in math involves checking if the equation or expression holds when the given values are substituted.
This is often an essential step in verifying solutions.

• If substituting ordered pairs into an equation results in the initial equation balancing correctly (both sides equal), the statement is true.
• Conversely, if the sides do not equal, the statement is false, as in the example where \(8 - 9\) did not equal \(-6\).

It's crucial to double-check these results, as any mistake might create an incorrect conclusion about the alignment of the ordered pair with the equation.

Substitution Method

The substitution method is widely used to test whether given values satisfy an equation. It involves replacing variables with specific numbers to see if a statement is valid.

• This technique is not only applicable for checking solutions but is instrumental in solving equations.
• For the exercise given, you substitute \(x = 3\) and \(y = 4\) into the equation, resulting in the expression \(2*4 - 3*3 = -1\).

By following the steps correctly, any discrepancy from the expected result indicates whether the initial statement was true or false. This hands-on approach gives clarity and surety about the correctness of mathematical relationships.