Problem 43 - Elementary Algebra

Question

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing. $$ \left\\{\begin{array}{l} x=-3 y+4 \\ 2 x+6 y=8 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \(2, \frac{2}{3}\).

Step 1: Rewrite each equation in slope-intercept form

Rewrite both equations to the form \(y = mx + b\). The first equation is already in a simple form, so let's rearrange the second equation. \ The first equation is \ x = -3y + 4 \.The second equation is \ 2x + 6y = 8 \. To write it in slope-intercept form, solve for \ y \: First, solve for \ 6y \: \ 6y = -2x + 8 \.Divide both sides by 6 to isolate \ y \: \ y = -\frac{1}{3}x + \frac{4}{3} \.

Step 2: Graph both equations

Graph the equations on the same coordinate plane:For the first equation \(x = -3y + 4\):- To find the points, we can create a table.- If \(y = 0\), then \ x = 4 \.- If \(y = 1\), then \ x = 1 \.- If \(y = 2\), then \ x = -2 \.For the second equation \ y = -\frac{1}{3}x + \frac{4}{3} \:- To find the points, we can create a table.- If \(x = 0\), then \ y = \frac{4}{3} \.- If \(x = 3\), then \ y = \frac{1}{3} \.- If \(x = -3\), then \ y = 2 \.Graph these points and then draw a line through them for each equation.

Step 3: Identify the point of intersection

The solution to the system of equations is the point where the two lines intersect. From the graphs created in Step 2, identify the coordinates where the two lines meet.

Step 4: Verify the solution

Substitute the coordinates of the intersection point back into the original equations to verify the solution. Both equations should be satisfied with these coordinates.

Key Concepts

Graphing Equations

Graphing is a way to visualize mathematical equations on a coordinate plane. When graphing equations, each equation is represented as a line. Every point on the line satisfies the equation. To graph an equation, you need points. You can get points by plugging in values for x or y and solving for the other variable. Plot these points on the coordinate plane then draw a line through them. This line represents all the solutions to the equation.

Slope-Intercept Form

In slope-intercept form, an equation of a line is written as \(y = mx + b\).
This form is useful because it clearly shows the slope \(m\) and the y-intercept \(b\). The slope ( \(m\) ) describes how steep the line is. It tells you how much y changes for a change in x. The y-intercept ( \(b\) ) is where the line crosses the y-axis, meaning it's the value of y when x is zero. Converting equations to this form makes them easier to graph because you know a starting point and how the line moves.

Solving Systems by Graphing

A system of linear equations consists of two or more linear equations with the same variables. Solving systems by graphing involves graphing each equation on the same set of axes. The solution is the point where the lines intersect. This point will satisfy all the equations in the system.
Steps to solve by graphing:

• Convert equations to slope-intercept form \(y = mx + b\) if necessary.
• Graph each equation on the same coordinate plane.
• Find the intersection point of the lines.
  

Remember, the intersection point is where the x and y values make all equations true.

Intersection Point Verification

After graphing the equations and finding the intersection point, you need to verify the solution. This step ensures the intersection point is correct and satisfies both equations.
To verify:

• Take the coordinates of the intersection point.
• Substitute them back into the original equations.
• Check if both equations are satisfied with these coordinates.
  

If both equations are true when you substitute the coordinates, the solution is correct. This helps confirm your graphical solution and ensures no mistakes were made while graphing or finding the point.