Problem 34 - Algebra 1

Question

Graph and check to solve the linear system. $$ \begin{aligned} &2.8 x+1.4 y=1.4\\\ &0.7 x-0.7 y=1.4 \end{aligned} $$

Step-by-Step Solution

Answer

The solution to the given system of linear equations is the point (0.33, -1.67).

Step 1: Rewrite in slope-intercept form

First, rewrite both equations in slope-intercept form (y = mx + c). Equation 1 becomes \(y = -2x + 1\) and Equation 2 becomes \(y = x - 2\).

Step 2: Plot the equations

Using any graphing tool, each equation can be plotted. The plot for Equation 1 will be a line passing through the points (0,1) with a slope of -2 while the plot for Equation 2 will be a line passing through (0,-2) and has a slope of 1.

Step 3: Find the point of intersection

The point where the two lines intersect on the graph is the solution to the system of linear equations. In this case, it's the point (0.33, -1.67).

Key Concepts

Slope-Intercept Form

When working with linear equations, the slope-intercept form is a very handy tool. This form is written as \( y = mx + c \), where \( m \) represents the slope and \( c \) represents the y-intercept.
This formula allows us to quickly identify how the line behaves and where it crosses the y-axis.
The **slope** \( m \) describes how steep the line is. A positive slope means the line ascends as it moves from left to right, while a negative slope means the line descends. The **y-intercept** \( c \) is the point where the line crosses the y-axis.
In our example, we transformed the equations:

• **Equation 1**: from \(2.8x + 1.4y = 1.4\) to \(y = -2x + 1\)
• **Equation 2**: from \(0.7x - 0.7y = 1.4\) to \(y = x - 2\)

This form makes it immensely clear how you'll plot each line.

Graphing Linear Systems

Graphing linear systems involves plotting each equation on the same graph to visually find the solution point. Each linear equation will appear as a line on this graph.
This method gives an immediate visual representation of where the lines intersect, helping us find solutions easier than with numerous calculations.**Plotting Lines:** You'll need a graphing tool or graph paper to plot these equations.

• **Equation 1**: The line passes through the y-intercept \((0, 1)\) with a slope of \(-2\). This means for every unit the line goes right (along the x-axis), it drops 2 units (along the y-axis).
• **Equation 2**: The line passes through the y-intercept \((0, -2)\) and goes up 1 unit for every 1 unit it goes right, as the slope is \(1\).

By following these plots, we can clearly see where the two lines intersect at a single point.

Solution of Linear Systems

The solution of a system of linear equations is typically the point where the lines intersect on a graph. This intersection point is the set of \(x, y\) values that satisfy both equations simultaneously.
**Intersection Point:**
In our example, we see that the lines intersect at the point \((0.33, -1.67)\). This means that if you substitute \(x = 0.33\) and \(y = -1.67\) into either equation from the system, both equations will hold true.
Understanding the intersection point as a solution is key:

• **It's where both equations are satisfied.**
• **No other points (within this context) will fulfill both equations at once.**

This method quickly demonstrates whether a system has a single solution (one intersection), no solution (parallel, non-intersecting lines), or infinitely many solutions (identical lines). By graphing the system, you gain a clear and direct insight into the relationships depicted by these linear equations.