Systems of Equations - SAT Math

$(x, y) = (6, 4)$. Add equations to get $2x = 12$, so $x = 6$, then $y = 4$.

Add/subtract equations to eliminate a variable. Multiply equations to make coefficients opposites, then combine.

$(x, y) = (-\frac{2}{3}, \frac{5}{3})$. Set equal: $2x + 3 = -x + 1$, solve to get $x = -\frac{2}{3}$.

$(x, y) = (0, 0)$. Add equations: $2x = 0$, so $x = 0$, then $y = 0$.

$x = 4$. Substitute $y = 0$ into first equation: $3x = 12$.

$(x, y) = (6, 1)$. Add equations to eliminate $y$: $2x = 12$, so $x = 6$.

Same slope and same intercept. Lines overlap completely when all coefficients are proportional.

Infinitely many solutions. Second equation is twice the first, so lines are identical.

$x = 4$. Substitute $y = 0$ into equation: $5x = 20$.

$(x, y) = (1, 3)$. Set equal: $x + 2 = -2x + 5$, solve to get $x = 1$.

Infinitely many solutions. Dependent equations represent the same geometric line.

Infinitely many solutions. Second equation is twice the first, creating coincident lines.

$x = 6$. Substitute $y = 0$ into equation: $2x = 12$.

$y = 3$. Substitute $x = 5$ into first equation directly.

Solve one equation for a variable, substitute in the other. Express one variable in terms of another from first equation.

$(x, y) = (3, 0)$. Add equations to eliminate $y$: $6x = 18$, so $x = 3$.

Lines are parallel, same slope, different intercepts. Lines never intersect when they have identical slopes.

Infinitely many solutions. Second equation is twice the first, creating identical lines.

Infinitely many solutions. Second equation is twice the first, so lines coincide.

Infinitely many solutions. Second equation is twice the first, creating identical lines.

Use matrices and row operations or inverses. Convert system to matrix form and solve systematically.

Infinitely many solutions. Second equation is $-2$ times the first, so lines coincide.

Determinant $\neq 0$. Non-zero determinant ensures coefficient matrix is invertible.

$(x, y) = (3, 2)$. Add equations to eliminate $y$: $2x = 6$, so $x = 3$.

Equations represent the same line. One equation is a multiple of the other.

$(x, y) = (2, 1)$. Add equations to eliminate $y$: $3x = 6$, so $x = 2$.

Plot equations, solution is intersection point. Draw both lines; they meet at one point.