Solve this system of equations using the substitution method.
[tex]
\begin{array}{l}
y=5 x-3 \
y=-6 x+8
\end{array}
[/tex]
Because both equations equal [tex]y[/tex], we can substitute one of the equations for [tex]y[/tex]. This will create an equation with only one variable, [tex]x[/tex].
[tex]
\begin{aligned}
5 x-3 & =-6 x+8 \
{[?] x-3 } & =8
\end{aligned}
[/tex]
Answer (2)
Substitute one equation for y into the other equation: 5 x − 3 = − 6 x + 8 .
Add 6 x to both sides: 11 x − 3 = 8 .
Add 3 to both sides: 11 x = 11 .
Divide by 11 : x = 1 . Substitute x = 1 into y = 5 x − 3 to get y = 2 . The solution is x = 1 , y = 2 .
Explanation
Setting up the problem We are given a system of two equations:
Equation 1: y = 5 x − 3 Equation 2: y = − 6 x + 8
Our goal is to solve this system using the substitution method. Since both equations are already solved for y , we can set them equal to each other. This will create an equation with only one variable, x .
Substitution Since both equations are equal to y , we can substitute one equation for y in the other equation. In this case, we set the two equations equal to each other:
5 x − 3 = − 6 x + 8
Adding 6x to both sides Now, we want to isolate x on one side of the equation. To do this, we first add 6 x to both sides of the equation:
5 x + 6 x − 3 = − 6 x + 6 x + 8
This simplifies to:
11 x − 3 = 8
Adding 3 to both sides Next, we add 3 to both sides of the equation:
11 x − 3 + 3 = 8 + 3
This simplifies to:
11 x = 11
Dividing by 11 Now, we divide both sides of the equation by 11 to solve for x :
11 11 x = 11 11
This gives us:
x = 1
Solving for y Now that we have the value of x , we can substitute it back into either of the original equations to solve for y . Let's use the first equation:
y = 5 x − 3
Substitute x = 1 :
y = 5 ( 1 ) − 3
y = 5 − 3
y = 2
Verification So, the solution to the system of equations is x = 1 and y = 2 . We can verify this solution by substituting these values into both original equations:
Equation 1: y = 5 x − 3 2 = 5 ( 1 ) − 3 2 = 5 − 3 2 = 2 (True)
Equation 2: y = − 6 x + 8 2 = − 6 ( 1 ) + 8 2 = − 6 + 8 2 = 2 (True)
Since the solution satisfies both equations, it is correct.
Final Answer Therefore, the solution to the system of equations is x = 1 and y = 2 .
Examples
Systems of equations are used in many real-world applications. For example, they can be used to model supply and demand in economics, where the intersection of the supply and demand curves represents the equilibrium price and quantity. They are also used in engineering to analyze circuits and structures, and in computer graphics to solve for the intersection of lines and planes. Understanding how to solve systems of equations is a fundamental skill in many fields.
By substituting y from one equation into the other, we isolate x , solve for it, and then substitute back to find y . The solution to the system of equations is x = 1 and y = 2 . We verified the solution by substituting back into both original equations and confirming they hold true.
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