91Ó°ÊÓ

To begin solving the equation, eliminate any fractions. This makes further steps easier. In our exercise, we have the equation: \[ y + 4 = \frac{5}{8}(x - 12) \]First, distribute the fraction to get rid of it. This involves multiplying each term inside the parentheses by the fraction: \[ y + 4 = \frac{5}{8}x - \frac{5}{8} \times 12 \]After simplifying, we get:\[ y + 4 = \frac{5}{8}x - 7.5 \]Now, the fraction is cleared, and we can move on to rearranging the equation into the slope-intercept form.

After clearing fractions, our goal is to write the equation in slope-intercept form (\[ y = mx + b \]). Here, \(m\) represents the slope. To isolate y, we subtract 4 from both sides of the equation. This gives us: \[ y = \frac{5}{8}x - 7.5 - 4 \]Simplifying, we get:\[ y = \frac{5}{8}x - 11.5 \]In this form, it's easy to identify the slope. The coefficient of \(x\) in this equation is \(\frac{5}{8}\), which is our slope \(m\).

Intercepts are the points where the line crosses the \(x\)-axis and \(y\)-axis. The \(y\)-intercept occurs where the line crosses the \(y\)-axis, which can be read directly from the slope-intercept form (\[ y = mx + b \]).- For the given equation \( y = \frac{5}{8}x - 11.5 \), the y-intercept \(b\) is \(-11.5\).To write it as a coordinate, we have:\[ (0, -11.5) \]- To find the \(x\)-intercept, set \(y\) to 0 and solve for \(x\). We have:\[ 0 = \frac{5}{8}x - 11.5 \]Solving for \(x\) requires adding 11.5 to both sides and multiplying by the reciprocal of \(\frac{5}{8}\):\[ 11.5 = \frac{5}{8}x \]\[ x = 11.5 \times \frac{8}{5} \]Simplifying, we find \(x = 18.4\). Thus, the \(x\)-intercept is: \[ (18.4, 0) \]

A linear equation forms a straight line when graphed on a coordinate plane. It is typically written in the form, \(y = mx + b\), where:- \(m\) is the slope, which represents the steepness of the line.- \(b\) is the y-intercept, the point where the line crosses the y-axis.These equations model a constant rate of change. They can be used in various applications, including physics to describe motion, economics to model growth, and everyday problem-solving.By clearing fractions and isolating variables as demonstrated, we simplify complex equations into this straightforward format.

Solving for variables involves isolating them on one side of the equation. In our example, we needed to isolate \(y\) to transform the equation into slope-intercept form. Here are the essential steps:- Begin by clearing any fractions through distribution or multiplication.- Rearrange terms to isolate the desired variable.- Simplify by performing arithmetic operations like addition, subtraction, multiplication, or division.Using these steps, we took \(y + 4 = \frac{5}{8}x - 7.5\) and isolated \(y\) to find:\[ y = \frac{5}{8}x - 11.5 \] This method is crucial not only in algebra but also in more advanced mathematics and various practical applications.