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Chapter 5: Problem 43

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

Short Answer

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

Step by step solution

01

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} \.
02

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.
03

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.
04

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Caitlyn sells her drawings at the county fair. She wants to sell at least 60 drawings and has portraits and landscapes. She sells the portraits for \(\$ 15\) and the landscapes for \(\$ 10\). She needs to sell at least \(\$ 800\) worth of drawings in order to earn a profit. (a) Write a system of inequalities to model this situation. (b) Graph the system. (c) Will she make a profit if she sells 20 portraits and 35 landscapes? (d) Will she make a profit if she sells 50 portraits and 20 landscapes?

In the following exercises, translate to a system of equations and solve. Lucinda had a pocketful of dimes and quarters with a value of \(\$ \$ 6.20\). The number of dimes is eighteen more than three times the number of quarters. How many dimes and how many quarters does Lucinda have?

Determine the Number of Solutions of a Linear System Without graphing the following systems of equations, determine the number of solutions and then classify the system of equations. $$ \left\\{\begin{array}{l} y=2 x+3 \\ 2 x-y=-3 \end{array}\right. $$

Mark is attempting to build muscle mass and so he needs to eat at least an additional 80 grams of protein a day. A bottle of protein water costs \(\$ 3.20\) and a protein bar costs \(\$ 1.75 .\) The protein water supplies 27 grams of protein and the bar supplies 16 gram. If he has \(\$ 10\) dollars to spend (a) Write a system of inequalities to model this situation. (b) Graph the system. (c) Could he buy 3 bottles of protein water and 1 protein bar? (a) Could he buy no bottles of protein water and 5 protein bars?

Mitchell currently sells stoves for company \(A\) at a salary of \(\$ 12,000\) plus a \(\$ 150\) commission for each stove he sells. Company B offers him a position with a salary of \(\$ 24,000\) plus a \(\$ 50\) commission for each stove he sells. How many stoves would Mitchell need to sell for the options to be equal?
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