Graph this system of equations:
[tex]
\begin{array}{l}
1.15 x+0.65 y=8.90 \\
x-3 y=-15
\end{array}
[/tex]
Answer (2)
To graph the system of equations, we convert each equation to slope-intercept form and plot them on a coordinate plane. The intersection point of the lines indicates the solution to the system. We can use y-intercepts and slopes to find and draw additional points on each line.
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Rewrite the first equation in slope-intercept form: y = − 13 23 x + 13 178 .
Rewrite the second equation in slope-intercept form: y = 3 1 x + 5 .
Plot the y-intercepts and use the slopes to find additional points on each line.
Draw the lines on the coordinate plane to visualize the system of equations.
Explanation
Analyze the problem We are given a system of two linear equations:
1.15 x + 0.65 y = 8.90 x − 3 y = − 15
Our objective is to graph this system of equations. To do this, we will rewrite each equation in slope-intercept form ( y = m x + b ), where m is the slope and b is the y-intercept.
Rewrite the first equation First, let's rewrite the first equation, 1.15 x + 0.65 y = 8.90 , in slope-intercept form. We solve for y :
0.65 y = − 1.15 x + 8.90
y = 0.65 − 1.15 x + 0.65 8.90
y = − 65 115 x + 65 890
y = − 13 23 x + 13 178
So, the slope of the first equation is − 13 23 ≈ − 1.769 and the y-intercept is 13 178 ≈ 13.692 .
Rewrite the second equation Now, let's rewrite the second equation, x − 3 y = − 15 , in slope-intercept form. We solve for y :
− 3 y = − x − 15
y = 3 1 x + 5
So, the slope of the second equation is 3 1 ≈ 0.333 and the y-intercept is 5 .
Graphing the equations Now we have both equations in slope-intercept form:
y = − 13 23 x + 13 178
y = 3 1 x + 5
To graph these lines, we plot the y-intercept for each equation on the coordinate plane. Then, using the slope, we find at least one additional point on each line. For the first equation, we start at ( 0 , 13 178 ) and use the slope − 13 23 to find another point. For example, we can go 13 units to the right and 23 units down. For the second equation, we start at ( 0 , 5 ) and use the slope 3 1 to find another point. For example, we can go 3 units to the right and 1 unit up. Finally, we draw a straight line through the points for each equation.
The graph The graph of the system is the two lines plotted on the same coordinate plane. The intersection point of the two lines is the solution to the system of equations.
Examples
Systems of equations are used in various real-life scenarios, such as determining the break-even point for a business. For example, if a company has fixed costs and variable costs, and they sell their product at a certain price, we can set up a system of equations to find the number of units they need to sell to cover their costs and start making a profit. Another example is in physics, where systems of equations can be used to analyze the forces acting on an object in equilibrium. These equations help engineers design structures that can withstand different loads and stresses.
