Use an Ilagram to work out the solution to these simultaneous equations:
[tex]
\begin{array}{l}
y=-2 x+4 \
y=\frac{1}{2} x+1
\end{array}
[/tex]
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Answer (2)
Set the two equations equal to each other: − 2 x + 4 = 2 1 x + 1 .
Solve for x : x = 5 6 .
Substitute the value of x into one of the equations to solve for y : y = 5 8 .
The solution to the simultaneous equations is x = 5 6 , y = 5 8 .
Explanation
Understanding the Problem We are given two simultaneous equations:
y = − 2 x + 4
y = 2 1 x + 1
We need to find the values of x and y that satisfy both equations.
Setting the Equations Equal Since both equations are solved for y , we can set them equal to each other to solve for x :
− 2 x + 4 = 2 1 x + 1
Solving for x Now, let's solve for x . First, add 2 x to both sides of the equation:
4 = 2 1 x + 2 x + 1
4 = 2 5 x + 1
Next, subtract 1 from both sides:
3 = 2 5 x
Finally, multiply both sides by 5 2 to isolate x :
x = 5 2 × 3
x = 5 6
Solving for y Now that we have the value of x , we can substitute it into either equation to solve for y . Let's use the second equation:
y = 2 1 x + 1
Substitute x = 5 6 :
y = 2 1 × 5 6 + 1
y = 5 3 + 1
y = 5 3 + 5 5
y = 5 8
Final Answer Therefore, the solution to the simultaneous equations is x = 5 6 and y = 5 8 .
Examples
Simultaneous equations are used in various real-world applications, such as determining the break-even point for a business. For example, if you have a cost function and a revenue function, setting them equal to each other (solving the simultaneous equations) will give you the point where your business starts making a profit. They are also used in physics to solve problems involving multiple forces or constraints, and in computer graphics to calculate intersections and transformations.
To solve the equations y = − 2 x + 4 and y = 2 1 x + 1 , first set them equal to find x = 5 6 . Substituting this value into one of the equations gives y = 5 8 . Thus, the solution is ( 5 6 , 5 8 ) .
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