Problem 29 - Algebra 1

Question

Graph and check to solve the linear system. $$ \begin{aligned} &15 x-10 y=-80\\\ &6 x+8 y=-80 \end{aligned} $$

Step-by-Step Solution

Answer

The solution to the system of equations is \((-4,2)\).

Step 1: Rewrite the Equations In Slope-Intercept Form

First, each equation should be solved for 'y'. Here's how it goes: for the first equation \(15x - 10y = -80\), it can be rewritten as \(y = 1.5x + 8\). Similarly, the second equation \(6x + 8y = -80\) can be rearranged as \(y = -0.75x - 10\).

Step 2: Graph the Equations

After turning each equation to slope-intercept form, the next step is to graph them. You'll get two lines. One line for each equation.

Step 3: Find the Intersection

The solution to the system is the coordinates of the point where the two lines intersect. In this case, the lines intersect at point \((-4,2)\).

Key Concepts

Graphing Linear Equations

Graphing linear equations helps us to see the relationship between variables visually. A linear equation creates a straight line on a graph. To graph a linear equation, you first need a set of axes, an x-axis, and a y-axis. Once these are set, you plot points that satisfy the equation and connect them to form a line.
There are different forms of equations that describe lines, such as slope-intercept form and point-slope form. Here, the graph of each equation in the linear system is a key tool. By graphing the equations, the point where they cross, if they do, shows the solution to the system.
Graphing is a powerful visual method that provides not only the solution but also insights into how linear equations behave, intersect, and can be parallel or coincidental.

Slope-Intercept Form

The slope-intercept form is a way of writing linear equations as: \[ y = mx + b \]here, \( m \) represents the slope of the line and \( b \) is the y-intercept. This form is particularly useful because it directly tells us about the slope and position of the line.
The slope, \( m \), shows how steep the line is. For example, a positive slope means the line rises as it goes from left to right, while a negative slope means it falls. The y-intercept, \( b \), tells us where the line crosses the y-axis.
For the equation \( 15x - 10y = -80 \), rewriting in slope-intercept form gives \( y = 1.5x + 8 \). This means the slope is 1.5, rising as it moves right, and it crosses the y-axis at 8. The second equation \( 6x + 8y = -80 \) becomes \( y = -0.75x - 10 \), indicating a negative slope, falling left to right, and a y-intercept at -10. Converting to slope-intercept form often makes graphing more straightforward.

Solving by Intersection

Finding a solution to a system of linear equations by looking at the intersection point is a straightforward concept. This method involves graphing both equations on the same axes. When we graph them, check where the lines intersect. This point is where both equations are true simultaneously, meaning it is the solution to this system.
In our example, we converted the equations to slope-intercept form and plotted them. By graphing the equations \( y = 1.5x + 8 \) and \( y = -0.75x - 10 \), we discovered they intersect at \((-4, 2)\), showing that \( x = -4 \) and \( y = 2 \) satisfy both equations.
This graphical tool gives an immediate visual confirmation of whether and where solutions exist, making it incredibly useful for linear systems.