Problem 31 - Algebra 1

Question

Graph and check to solve the linear system. $$ \begin{aligned} &x-8 y=-40\\\ &-5 x+8 y=8 \end{aligned} $$

Step-by-Step Solution

Answer

The solution to the system of equations is (4, 5.5).

Step 1: Reformat equations

First, you need to reformat the equations to slope-intercept form \(y = mx + b\). For the first equation, solve for y: \(x - 8y = -40\), or equivalently, \(8y = x + 40\). Thus, \(y = \frac{1}{8}x + 5\). For the second equation, solve for y: \(-5x + 8y = 8\), or equivalently, \(8y = 5x + 8\). Thus, \(y = \frac{5}{8}x + 1\).

Step 2: Graph the equations

With the equations in slope-intercept form, \((y = mx + b)\), you can plot each one on the same graph. For \(y = \frac{1}{8}x + 5\), the y-intercept (b) is 5 and the slope (m) is \frac{1}{8}. So you start at 5 on the y-axis (the y-intercept), and then you go right one unit and up \(\frac{1}{8}\) of a unit, continuing for a few more points. Next, for \(y = \frac{5}{8}x + 1\), the y-intercept is 1 and the slope is \frac{5}{8}. Again start at 1 on the y-axis (the y-intercept), this time going right one unit and up \(\frac{5}{8}\) of a unit for a few points. Join the points for each equation to create two lines.

Step 3: Find the intersection point

The graphs of the two distinct linear equations represent lines in a plane. The point where these two lines intersect (if it exists) is the solution to the system of equations. In this case, by observing the graph, you should find that they intersect at some point. To verify, solve the system of equations using substitution or elimination. Set the two equations equal to each other to get \(\frac{1}{8}x + 5 = \frac{5}{8}x + 1\). Solve for x, which results in \(x = 4\). Substitute \(x = 4\) into either one of the original equations, for example \(x - 8y = -40\), and solve for y. So \(4 - 8y = -40\) gives \(y = 5.5\). Hence they intersect at (4, 5.5).

Step 4: Check the solution

Check if (4, 5.5) is a solution to the system of equations by substituting these values into the original equations. Both should be satisfied, which confirms that the lines indeed cross at (4, 5.5). Hence (4, 5.5) is a solution to the system of equations.

Key Concepts

Slope-Intercept Form

When it comes to graphing linear equations, the slope-intercept form is a critical player. This form is written as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) is the y-intercept. The y-intercept is where the line crosses the y-axis and the slope indicates the steepness of the line, as well as its direction.

Understanding how to convert a linear equation to slope-intercept form is essential as it facilitates easier graphing. To illustrate, let's consider an equation \( x - 8y = -40 \). To solve for y, and thus get it into slope-intercept form, you'd rearrange the equation to yield \( y = \frac{1}{8}x + 5 \). Here, \( \frac{1}{8} \) is the slope and 5 is the y-intercept. This transformation allows for quick graphing, as all you'll need to start is the y-intercept, and then you can use the slope to find other points on the line.

Graphing Linear Equations

Graphing linear equations involves translating algebraic equations into visual lines on a graph. Each line's starting point is found at its y-intercept (the value of \( y \) when \( x = 0 \)), as we saw with the slope-intercept form.

For instance, with \( y = \frac{1}{8}x + 5 \), you would begin at the point (0, 5) on the graph. Then, apply the slope - rise over run. With a slope of \( \frac{1}{8} \), for every one unit you move horizontally to the right, you move upward by \( \frac{1}{8} \) units. Plot a few points this way, draw a line through them, and you've graphed the equation. This method ensures accurate representation of the equation on a graph, making it easier to analyze the relationship it describes.

System of Equations

A system of equations is simply a set of two or more equations that you deal with together, often because you want to find where they intersect, or to figure out the values of variables they share. You may encounter a system like:
\[ \begin{aligned}&x - 8 y = -40\ &-5 x + 8 y = 8\end{aligned} \]
Graphing each equation from this system on the same coordinate plane can help you visually identify their point of intersection, which signifies the solution to the system. By graphing, you convert an abstract algebraic concept into a concrete visual representation, which is a powerful method for better understanding the relationships put forward by the system.

Intersection Point of Lines

The intersection point of lines is where two or more lines on a graph cross each other. This point is of great significance when you are working with a system of equations, as it represents the set of coordinates that satisfy all equations in the system simultaneously.

To find this point, you rely on your ability to graph the lines as discussed earlier. When you graph the two equations \( y = \frac{1}{8}x + 5 \) and \( y = \frac{5}{8}x + 1 \), their paths will cross at exactly one point if they are not parallel. This coordinates of this point are the 'solution' to the system. Algebraically, you can find it by setting the two equations equal to each other and solving for \( x \) and \( y \). In the given exercise, after graphing the two lines, the intersection point was determined to be (4, 5.5), which effectively means that both lines satisfy the equation \( x - 8 y = -40 \) and \( -5 x + 8 y = 8 \) when \( x = 4 \) and \( y = 5.5 \).