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Solve System of Linear Equations with Substitution: $y=-3x+5$, $5x-4y=-3$
Mathematics
Grade 8 (Junior High School)
Question Content
Solve using substitution, write solution as (x,y) no spaces. \n$y = -3x + 5$\n$5x - 4y = -3$
Correct Answer
(1,2)
Detailed Solution Steps
1
Step 1: Substitute $y=-3x+5$ from the first equation into the second equation $5x-4y=-3$, resulting in: $5x - 4(-3x + 5) = -3$
2
Step 2: Expand and simplify the equation: $5x + 12x - 20 = -3$. Combine like terms: $17x - 20 = -3$
3
Step 3: Isolate $x$: Add 20 to both sides: $17x = -3 + 20 = 17$. Divide both sides by 17: $x=1$
4
Step 4: Substitute $x=1$ back into $y=-3x+5$: $y=-3(1)+5=-3+5=2$
5
Step 5: Write the solution in the required (x,y) no-spaces format: (1,2)
Knowledge Points Involved
1
Substitution Method for Linear Systems
A standard algebraic method for solving systems of linear equations where one variable is expressed in terms of the other, then substituted into the second equation to solve for the remaining variable. It works for consistent, independent systems with exactly one solution.
2
Distributive Property
A property of multiplication that states $a(b+c)=ab+ac$. Used here to expand $-4(-3x+5)$ into $12x-20$, which is essential for simplifying the substituted equation.
3
Verifying Solutions to Linear Systems
After finding values for x and y, you can check by plugging them back into both original equations to ensure they satisfy each equation. For this problem, $2=-3(1)+5$ and $5(1)-4(2)=5-8=-3$, confirming the solution is correct.
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