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)
By converting both equations to slope-intercept form, we can find their slopes and y-intercepts, allowing for accurate graphing. Plot the lines based on the intercepts and slopes, and their intersection point will provide the solution to the system of equations.
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Convert both equations to slope-intercept form: y = − 1.769 x + 13.692 and y = 3 1 x + 5 .
Identify the slopes and y-intercepts for each equation.
Plot the y-intercepts on the y-axis and use the slopes to find additional points on each line.
Draw the lines through the points to graph the system of equations. The intersection point represents the solution. The graph of the system of equations
Explanation
Understanding 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 first convert each equation into slope-intercept form ( y = m x + b ), where m is the slope and b is the y-intercept. Then, we can plot the lines on a graph using their slopes and y-intercepts.
Converting the First Equation Let's convert the first equation, 1.15 x + 0.65 y = 8.90 , to slope-intercept form. First, isolate the term with y :
0.65 y = − 1.15 x + 8.90
Now, divide both sides by 0.65 :
y = 0.65 − 1.15 x + 0.65 8.90
Calculating the values, we get:
y = − 1.769 x + 13.692 (approximately)
Converting the Second Equation Now, let's convert the second equation, x − 3 y = − 15 , to slope-intercept form. First, isolate the term with y :
− 3 y = − x − 15
Now, divide both sides by − 3 :
y = 3 1 x + 5
Identifying Slopes and Y-Intercepts Now we have both equations in slope-intercept form:
y = − 1.769 x + 13.692
y = 3 1 x + 5
From these equations, we can identify the slopes and y-intercepts:
Equation 1: slope m 1 = − 1.769 , y-intercept b 1 = 13.692
Equation 2: slope m 2 = 3 1 = 0.333 , y-intercept b 2 = 5
Plotting the Lines To graph the lines, we can plot the y-intercepts on the y-axis and use the slopes to find another point on each line. For example:
Equation 1: Start at ( 0 , 13.692 ) . A slope of − 1.769 means for every 1 unit increase in x , y decreases by 1.769 . So, another point could be ( 1 , 13.692 − 1.769 ) = ( 1 , 11.923 ) .
Equation 2: Start at ( 0 , 5 ) . A slope of 3 1 means for every 3 units increase in x , y increases by 1 . So, another point could be ( 3 , 5 + 1 ) = ( 3 , 6 ) .
Plot these points and draw the lines through them.
Final Answer The system of equations is now graphed by plotting the two lines. The intersection point of the two lines represents the solution to the system of equations.
Examples
Systems of equations are incredibly useful in real life. Imagine you're trying to figure out the break-even point for a new business venture. You can model your costs with one equation and your revenue with another. The point where the lines intersect (the solution to the system) tells you exactly how much you need to sell to cover your costs! This is just one example; systems of equations pop up in physics, engineering, economics, and many other fields.
