From the first equation, solve for x in terms of y:
2x = 9 - 5y
x = (9 - 5y)/2
Substitute the value of x into the second equation:
2((9 - 5y)/2) + 3y = 7
9 - 5y + 3y = 7
-2y = -2
y = 1
Substitute the value of y back into the first equation to solve for x:
2x + 5(1) = 9
2x + 5 = 9
2x = 4
x = 2
Therefore, the solution to the system of equations is x = 2 and y = 1, indicating the point of intersection for the two given lines in the coordinate plane.
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USING GRAPHICAL METHODS
We can plot the two equations on a graph and find the point of intersection, which represents the solution. Here are the steps to solve it graphically:
Let’s rewrite the equations in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept:
Equation 1: 2x + 5y = 9
Rewrite it as: y = (-2/5)x + 9/5
Equation 2: 2x + 3y = 7
Rewrite it as: y = (-2/3)x + 7/3
Plot the two lines on a coordinate plane using the slopes and y-intercepts.
For the first equation, the slope is -2/5 and the y-intercept is 9/5. Plot a point at (0, 9/5) and use the slope to find additional points. For example, if x = 5, then y = (-2/5)(5) + 9/5 = 1. Plot another point at (5, 1).
For the second equation, the slope is -2/3 and the y-intercept is 7/3. Plot a point at (0, 7/3) and find additional points using the slope. For example, if x = 3, then y = (-2/3)(3) + 7/3 = 1. Plot another point at (3, 1).
Identify the point of intersection of the two lines. This point represents the solution to the system of equations.
By visually inspecting the graph, you can see that the two lines intersect at the point (2, 1). This means that x = 2 and y = 1 satisfy both equations.
Therefore, the solution to the system of equations is x = 2 and y = 1.
1. Hello let's solve the system of equations 2x + 5y = 9 and 2x + 3y = 7 using algebraic methods
Equation 1: 2x + 5y = 9
Equation 2: 2x + 3y = 7
Step 1: Let's solve Equation 1 for x.
We isolate x by subtracting 5y from both sides:
2x = 9 - 5y
Divide both sides by 2 to solve for x:
x = (9 - 5y) / 2
Step 2: Substitute x in terms of y into Equation 2.
Replace x in Equation 2 with (9 - 5y) / 2:
2((9 - 5y) / 2) + 3y = 7
Simplify the equation:
(9 - 5y) + 3y = 7
Combine like terms:
9 - 2y = 7
Subtract 9 from both sides:
-2y = -2
Divide by -2:
y = 1
Step 3: Substitute y = 1 back into Equation 1 to solve for x.
Replace y with 1 in Equation 1:
2x + 5(1) = 9
2x + 5 = 9
Subtract 5 from both sides:
2x = 4
Divide by 2:
x = 2
So, our solution is x = 2 and y = 1.
[tex] [/tex]
2. Now, let's solve the same system of equations using graphical methods:
Graphically, we will plot the equations 2x + 5y = 9 and 2x + 3y = 7 on a coordinate plane and determine the point of intersection.
For Equation 1: 2x + 5y = 9, we can rewrite it as:
y = (9 - 2x) / 5
For Equation 2: 2x + 3y = 7, we can rewrite it as:
y = (7 - 2x) / 3
Now, let's plot the lines on a graph and locate the point of intersection. By visually examining the graph, we can determine that the lines intersect at the point (2, 1), which gives us the solution to the system.
To summarize, the solution to the system of equations is x = 2 and y = 1, which we obtained through both algebraic and graphical methods.
Answers & Comments
USING ALGEBRAIC METHODS
Let’s solve it using the method of substitution:
From the first equation, solve for x in terms of y:
Substitute the value of x into the second equation:
Substitute the value of y back into the first equation to solve for x:
Therefore, the solution to the system of equations is x = 2 and y = 1, indicating the point of intersection for the two given lines in the coordinate plane.
[tex] \\ [/tex]
USING GRAPHICAL METHODS
We can plot the two equations on a graph and find the point of intersection, which represents the solution. Here are the steps to solve it graphically:
Let’s rewrite the equations in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept:
Rewrite it as: y = (-2/3)x + 7/3
Plot the two lines on a coordinate plane using the slopes and y-intercepts.
For the first equation, the slope is -2/5 and the y-intercept is 9/5. Plot a point at (0, 9/5) and use the slope to find additional points. For example, if x = 5, then y = (-2/5)(5) + 9/5 = 1. Plot another point at (5, 1).
For the second equation, the slope is -2/3 and the y-intercept is 7/3. Plot a point at (0, 7/3) and find additional points using the slope. For example, if x = 3, then y = (-2/3)(3) + 7/3 = 1. Plot another point at (3, 1).
Identify the point of intersection of the two lines. This point represents the solution to the system of equations.
By visually inspecting the graph, you can see that the two lines intersect at the point (2, 1). This means that x = 2 and y = 1 satisfy both equations.
Therefore, the solution to the system of equations is x = 2 and y = 1.
SOLUTION
1. Hello let's solve the system of equations 2x + 5y = 9 and 2x + 3y = 7 using algebraic methods
Step 1: Let's solve Equation 1 for x.
We isolate x by subtracting 5y from both sides:
Divide both sides by 2 to solve for x:
Step 2: Substitute x in terms of y into Equation 2.
Replace x in Equation 2 with (9 - 5y) / 2:
Simplify the equation:
Combine like terms:
Subtract 9 from both sides:
Divide by -2:
Step 3: Substitute y = 1 back into Equation 1 to solve for x.
Replace y with 1 in Equation 1:
Subtract 5 from both sides:
Divide by 2:
So, our solution is x = 2 and y = 1.
[tex] [/tex]
2. Now, let's solve the same system of equations using graphical methods:
Graphically, we will plot the equations 2x + 5y = 9 and 2x + 3y = 7 on a coordinate plane and determine the point of intersection.
For Equation 1: 2x + 5y = 9, we can rewrite it as:
For Equation 2: 2x + 3y = 7, we can rewrite it as:
Now, let's plot the lines on a graph and locate the point of intersection. By visually examining the graph, we can determine that the lines intersect at the point (2, 1), which gives us the solution to the system.
To summarize, the solution to the system of equations is x = 2 and y = 1, which we obtained through both algebraic and graphical methods.
[tex] \\ [/tex]
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