find the solution the the system of equations by using either graphing or substitution
Y = 4x and y + x = 5
Substitution method:
We can substitute the first equation into the second equation for y, giving us:
4x + x = 5
Simplifying:
5x = 5
x = 1
Now we can use the first equation to solve for y:
y = 4(1)
y = 4
Therefore, the solution to the system of equations is (1,4).
Graphing method:
We can graph each equation on the same coordinate axes and look for the intersection point:
y = 4x
y + x = 5
To graph y = 4x, we can plot the point (0,0) and use the slope of 4 to find additional points. We can also use the x-intercept (where y = 0) to find another point:
y = 4x
0 = 4x
x = 0
So the point (0,0) and (1,4) are on the first line.
To graph y + x = 5, we can rearrange to solve for y and graph the line using two points:
y = 5 - x
When x = 0, y = 5. When x = 5, y = 0. So the points (0,5) and (5,0) are on the second line.
Graphing the two lines, we see that they intersect at the point (1,4). This confirms our solution found using substitution.
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To find the solution to the system of equations, we can use substitution method.
We are given the equations:
1) y = 4x
2) y + x = 5
We can substitute equation 1) into equation 2) to solve for x:
Substitute y from equation 1) into equation 2):
4x + x = 5
Combine like terms:
5x = 5
Divide by 5:
x = 1
Now, substitute the value of x back into equation 1) to solve for y:
y = 4(1)
y = 4
Therefore, the solution to the system of equations is x = 1 and y = 4.
To solve the system of equations using graphing, follow these steps:
1. Rewrite both equations in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept.
Equation 1: Y = 4x
Equation 2: y + x = 5 (rewrite as y = -x + 5)
2. Plot the graphs of both equations on the same coordinate plane. To do this, select appropriate x-values, substitute them into the equations to find the corresponding y-values, and plot the points. Repeat this process for multiple values to obtain several points that lie on each line.
3. Connect the plotted points for each equation with a straight line.
4. The solution to the system of equations is the point where the two lines intersect. Read the coordinates of the intersection point.
Alternatively, to solve the system of equations using substitution, follow these steps:
1. Solve one of the equations for one variable in terms of the other variable. In this case, we can solve the second equation for y: y = 5 - x.
2. Substitute the expression obtained in step 1 into the other equation. Replace y in Equation 1 with (5 - x): Y = 4x becomes (5 - x) = 4x.
3. Solve the resulting equation for x by simplifying and isolating the variable. In this case, add x to both sides of the equation, resulting in: 5 = 5x. Then divide both sides by 5 to get x = 1.
4. Substitute the value of x back into one of the original equations to solve for y. Using Equation 2: y + x = 5, substitute x = 1, solving for y gives us y = 4.
5. The solution to the system of equations is the ordered pair (x, y). In this case, the solution is (1, 4).