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How to Solve the System of Linear Equations $\\begin{cases}3x + 2y = -26 \\\\ 4x - 5y = -4\\end{cases}$
Mathematics
Grade 8 (Junior High School)
Question Content
Solve the system of linear equations: $\\begin{cases}3x + 2y = -26 \\\\ 4x - 5y = -4\\end{cases}$
Correct Answer
$x=-6, y=-4$
Detailed Solution Steps
1
Step 1: Use the elimination method. Multiply the first equation by 5 and the second equation by 2 to make the coefficients of $y$ opposites: $5(3x+2y)=5\\times(-26) \\Rightarrow 15x+10y=-130$; $2(4x-5y)=2\\times(-4) \\Rightarrow 8x-10y=-8$
2
Step 2: Add the two new equations together to eliminate $y$: $(15x+10y)+(8x-10y)=-130+(-8) \\Rightarrow 23x=-138$
3
Step 3: Solve for $x$: $x=-138\\div23=-6$
4
Step 4: Substitute $x=-6$ into the first original equation $3x+2y=-26$: $3\\times(-6)+2y=-26 \\Rightarrow -18+2y=-26$
5
Step 5: Solve for $y$: $2y=-26+18=-8 \\Rightarrow y=-4$
6
Step 6: Verify by substituting $x=-6, y=-4$ into the second equation: $4\\times(-6)-5\\times(-4)=-24+20=-4$, which matches the right-hand side, so the solution is valid.
Knowledge Points Involved
1
System of Linear Equations in Two Variables
A set of two linear equations with two unknown variables, where the solution is an ordered pair $(x,y)$ that satisfies both equations simultaneously. It is commonly used to model real-world scenarios with two related unknown quantities.
2
Elimination Method for Solving Linear Systems
A method where you multiply one or both equations by a constant to make the coefficients of one variable opposites, then add the equations to eliminate that variable, solve for the remaining variable, and substitute back to find the other variable. It is a standard, efficient method for solving linear systems.
3
Substitution Method for Solving Linear Systems
An alternative method where you solve one equation for one variable in terms of the other, then substitute that expression into the second equation to get an equation with one variable, solve it, and substitute back. It is especially useful when one variable has a coefficient of 1 or -1.
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