Problem 12 - Elementary Algebra

Question

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing. $$ \left\\{\begin{array}{l} -2 x+3 y=-3 \\ x+y=4 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is (3, 1).

Step 1 - Write Equations in Slope-Intercept Form

Rewrite each equation in the form of y = mx + b. For the first equation: \[-2x + 3y = -3\] Add 2x to both sides: \[3y = 2x - 3\] Divide by 3: \[y = \frac{2}{3}x - 1\] For the second equation: \[x + y = 4\] Subtract x from both sides: \[y = -x + 4\]

Step 2 - Graph the First Equation

Graph y = \(\frac{2}{3}x - 1\) on a coordinate plane. Start by plotting the y-intercept at (0, -1). Then, use the slope \(\frac{2}{3}\) to find another point by going up 2 units and right 3 units.

Step 3 - Graph the Second Equation

Graph y = -x + 4 on the same coordinate plane. Start by plotting the y-intercept at (0, 4). Then, use the slope -1 to find another point by going down 1 unit and right 1 unit.

Step 4 - Find the Intersection Point

Identify the point where the two lines intersect. This point is the solution to the system of equations. Check the graph to see where the lines \(y = \frac{2}{3}x - 1\) and \(y = -x + 4\) intersect.

Step 5 - Verify the Solution

Verify the coordinates of the intersection point by substituting them back into the original equations to ensure they satisfy both equations.

Key Concepts

linear equations

Linear equations are mathematical expressions that represent straight lines when graphed on a coordinate plane. These equations are typically written in the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are variables. Linear equations have two main characteristics: the line is straight, and it has no exponents higher than 1 for any variable.
In solving systems of linear equations, you will often be working with more than one linear equation. The goal is to find a common solution that satisfies all the involved equations.

slope-intercept form

The slope-intercept form of a linear equation is \(y = mx + b\). In this format:

• 'm' represents the slope of the line
• 'b' represents the y-intercept, which is where the line crosses the y-axis.

For instance, consider the equation \(y = \frac{2}{3}x - 1\). Here, \(\frac{2}{3}\) is the slope, showing that for every 3 units you move to the right along the x-axis, you will move up 2 units along the y-axis. \(-1\) is the y-intercept, indicating that the line crosses the y-axis at -1.
Using the slope-intercept form is very helpful for graphing because it allows you to quickly identify the y-intercept and slope, enabling you to sketch the line more easily.

graphing techniques

Graphing linear equations is a visual way to find solutions. Here are some steps to help you:

• Start with the y-intercept: This is where the line crosses the y-axis.
• Use the slope to find other points: The slope 'm' tells you how to move from the y-intercept. If \(m = \frac{2}{3}\), go up 2 units and right 3 units from the y-intercept.
• Draw the line: Connect the points with a straight line.
• Repeat for additional equations: If dealing with a system of equations, graph the other lines on the same coordinate plane.

Accurate graphing is essential for solving systems of linear equations because the intersection point gives the solution. Remember to label points clearly and use a ruler for straight lines.

intersection point

The intersection point of two lines on a graph is where the lines cross. This point represents the common solution to both equations in a system of linear equations. To identify the intersection point:

• Graph each equation on the same coordinate plane.
• Look for the point where the lines meet.

For example, if the lines \(y = \frac{2}{3}x - 1\) and \(y = -x + 4\) intersect at point (3,1), then 3 is the value of x and 1 is the value of y that satisfy both equations. It's always a good idea to verify this by substituting back into the original equations to confirm they hold true.