Problem 78 - Introductory and Intermediate Algebra for College Students 4th

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 line whose equation is \(y-3=7(x+2)\) passes through \((-3,2)\)

Step-by-Step Solution

Answer

False. The given point \((-3,2)\) does not lie on the line \(y=7x + 17\). The corrected statement would be: The line whose equation is \(y=7x+17\) does not pass through \((-3,2)\).

Step 1: Identify the given point and equation of line

The coordinates of the point given are (-3, 2), and the equation of the line is \(y-3=7(x+2)\).

Step 2: Rearrange the Equation

Rewriting the given equation, we have \(y = 7x + 14 + 3\), which simplifies to \(y = 7x + 17\)

Step 3: Substitute the Point Into the Equation

Next, we will substitute x=-3 and y=2 into the equation \(y = 7x + 17\) which gives \(2 = 7*(-3) + 17\), and further simplifies to \(2=-21+17\)

Step 4: Evaluate and Conclude

Solving this gives \(2 \neq -4\), implying the point (-3,2) does not lie on the line.

Key Concepts

Point-Slope Form

The point-slope form is a handy way to write the equation of a line when you know its slope and a point on the line. The general format of this form is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope, and \( (x_1, y_1) \) are the coordinates of the given point.
It simplifies the process of finding a linear equation because once you have one point and the slope, you can plug them directly into the equation.
This form is particularly useful because:

• It is quick to derive the line's equation if the slope and one point are known.
• It is straightforward to manipulate into other forms like the slope-intercept form.

In the given exercise, we started with the point-slope form \( y-3=7(x+2) \). Here, 7 is the slope, and \(-2, 3\) is a point on the line. Notice how using this format allows checking whether other points lie on this line.

Equation of a Line

The equation of a line describes every point that lies on that line. There are multiple forms of the equation, such as point-slope form, standard form, or the most common slope-intercept form, which is written as \(y = mx + b\). In this form, \(m\) is the slope, and \(b\) is the y-intercept.
In our exercise, after rewriting the initial equation \( y-3=7(x+2) \) into slope-intercept form, we got \( y = 7x + 17 \). This means the line's slope is 7, and it crosses the y-axis at 17.
Understanding different forms of line equations is crucial because:

• They provide various insights, like the slope indicating steepness or direction.
• They allow checking if a point, like \((-3, 2)\), resides on the line.

Remember that converting between forms can help reveal both hidden points and slopes.

Substitution Method

The substitution method is a powerful algebraic technique used to verify whether a specific point lies on a line. By replacing the variables \(x\) and \(y\) with the coordinates of the point, you can see if the resulting statement holds true.
In our example, we used the line equation \( y = 7x + 17 \). By substituting \( x = -3 \) and \( y = 2 \), we calculated the expression \( 2 = 7(-3) + 17 \). This simplified to \( 2 = -21 + 17 \), which results in \( 2 eq -4 \).
This tells us the point \((-3, 2)\) is not on the line.

• Substitution helps in quickly verifying solutions.
• It is a straightforward check for whether points satisfy the given equation.

For any equation of a line, this method is a reliable and easy way to validate specific points.