Math Problem Statement

Which system has exactly one solution?

Solution

To determine which system of equations has exactly one solution, we need to check each pair of equations to see if their graphs intersect at a single point.

First system:

6x + 3y &= -1 \\ 2x + y &= 4 \end{aligned}$$ We can solve the second equation for $$y$$: $$y = 4 - 2x$$ Substitute this into the first equation: $$6x + 3(4 - 2x) = -1 \\ 6x + 12 - 6x = -1 \\ 12 = -1$$ This is a contradiction, meaning the system has no solution. ### Second system: $$\begin{aligned} y &= 4x - 5 \\ y &= -\frac{1}{4}x - 5 \end{aligned}$$ Since both equations are equal to $$y$$, set them equal to each other: $$4x - 5 = -\frac{1}{4}x - 5$$ Add 5 to both sides: $$4x = -\frac{1}{4}x$$ Multiply through by 4 to eliminate the fraction: $$16x = -x$$ Add $$x$$ to both sides: $$17x = 0 \implies x = 0$$ Substitute $$x = 0$$ into one of the equations to find $$y$$: $$y = 4(0) - 5 = -5$$ Thus, this system has exactly one solution: $$(0, -5)$$. ### Third system: $$\begin{aligned} y + 4x &= -2 \\ y &= -4x + 5 \end{aligned}$$ Substitute the second equation into the first: $$(-4x + 5) + 4x = -2 \\ 5 = -2$$ This is another contradiction, meaning this system has no solution. ### Fourth system: $$\begin{aligned} 3x - y &= 2 \\ y - 4 &= 3(x - 2) \end{aligned}$$ Simplify the second equation: $$y - 4 = 3x - 6 \implies y = 3x - 2$$ Substitute this into the first equation: $$3x - (3x - 2) = 2 \\ 3x - 3x + 2 = 2 \\ 2 = 2$$ This is true for all values of $$x$$, meaning the system has infinitely many solutions. ### Conclusion: The second system has exactly one solution. Would you like more details or have any questions? Here are 5 related questions to deepen your understanding: 1. How can you determine if a system has no solution graphically? 2. What does it mean for a system of equations to be inconsistent? 3. How do you find the point of intersection for two linear equations? 4. What is the significance of having infinitely many solutions? 5. How can substitution help solve a system of equations? **Tip**: When solving systems, always check if the lines are parallel, intersecting, or coincident for a quick understanding of the number of solutions.

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Math Problem Analysis

Mathematical Concepts

Systems of Linear Equations
Graphical Solutions
Substitution Method
Consistency of Equations

Formulas

Solving linear systems by substitution
Checking for one solution: intersecting lines

Theorems

Fundamental Theorem of Algebra for linear systems

Suitable Grade Level

Grades 8-10

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