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How to Solve the Linear System $\\begin{cases} y=3x+1 \\\\ 3y-4x=13 \\end{cases}$ Using Substitution Method
Mathematics
Grade 9 (Junior High School)
Question Content
Solve the system of equations by the substitution method. \n$\\begin{cases} y=3x+1 \\\\ 3y-4x=13 \\end{cases}$\nSelect the correct choice below and, if necessary, fill in the answer box to complete your choice.\nA. The solution is ____. (Simplify your answer. Type an ordered pair.)\nB. There are infinitely many solutions.\nC. There is no solution.
Correct Answer
A. The solution is (2, 7)
Detailed Solution Steps
1
Step 1: Identify the already isolated variable. The first equation $y=3x+1$ gives $y$ directly in terms of $x$, so we will substitute this expression for $y$ into the second equation.
2
Step 2: Substitute $y=3x+1$ into $3y-4x=13$. This gives the equation: $3(3x+1)-4x=13$.
3
Step 3: Simplify and solve for $x$. First expand the left side: $9x+3-4x=13$. Combine like terms: $5x+3=13$. Subtract 3 from both sides: $5x=10$. Divide both sides by 5: $x=2$.
4
Step 4: Substitute $x=2$ back into the first equation $y=3x+1$ to find $y$. Calculate $y=3(2)+1=6+1=7$.
5
Step 5: Verify the solution by plugging $x=2$ and $y=7$ into the second equation: $3(7)-4(2)=21-8=13$, which matches the right side. The solution is valid, so we choose option A and write the ordered pair (2, 7).
Knowledge Points Involved
1
Substitution Method for Systems of Linear Equations
This is a method to solve a system of two linear equations with two variables. It involves isolating one variable in one equation, then substituting that expression into the other equation to create a single-variable equation, which can be solved directly. It is most efficient when one equation already has a variable isolated (like $y=mx+b$).
2
Solving Linear Equations in One Variable
This involves using inverse operations (addition/subtraction, multiplication/division) to isolate the variable. Key steps include combining like terms, simplifying expressions, and performing the same operation on both sides of the equation to maintain equality.
3
Verifying Solutions to Systems of Equations
After finding a potential solution (ordered pair), substitute the values of $x$ and $y$ into both original equations. If both equations are satisfied (both sides are equal), the solution is valid.
4
Types of Solutions for Linear Systems
A consistent independent system (like this problem) has exactly one solution. A consistent dependent system has infinitely many solutions (the equations are equivalent lines). An inconsistent system has no solution (the lines are parallel and never intersect).
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